Chapters 3 and 6 Sample Problems Provide an appropriate response.

Transcription

Chapters 3 and 6 Sample Problems
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1) Suppose you were to collect data for the pair of given variables in order to make a scatterplot. Determine for
each variable if it is the explanatory variable or the response variable.
Variables: Cloudy days, rainy days
2) Suppose you were to collect data for the pair of given variables in order to make a scatterplot. Determine for
each variable if it is the explanatory variable or the response variable.
Variables: Minutes of homework, grade on exam
3) Hi-Tech Agi Inc wants to determine if the rainfall in inches can be used to predict the yield per acre on a corn
farm. Identify the response variable.
Fill in the blank.
4) The ____________________ defines the groups to be compared with respect to values on the response variable.
5) Whenever we distinguish between a response variable and an explanatory variable, it is natural to form
conditional proportions for categories of the ____________________.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Select the most appropriate answer.
6) The ____________________ is the outcome variable on which comparisons are made.
A) lurking variable
B) response variable
C) explanatory variable
D) predictor variable
E) Both b. and c.
7) For the following pairs of variables, which more naturally is the response variable and which is the explanatory
variable?
a.
b.
c.
College grade point average and college entrance exam score
Students' interest and ability in studying a foreign language
Speed of professional advancement and standard of dress
8) A large manufacturer hires many handicapped workers and keeps track of both their type of handicap and
their level of performance.
a.
b.
Identify the two variables.
Identify the response variable and the explanatory variable.
1
Complete the conditional proportion table and use it to solve the problem.
9) The partially filled conditional proportion table gives the relative frequencies of the data on age (in years) and
sex from the residents of a retirement home.
Age (yrs)
60-69
70-79
Over 79
Total
Male
0.19
0.1
0.11
Female
0.2
0.1
0.3
Total
1
What percentage of residents are males over 79?
10) The partially filled conditional proportion table gives the relative frequencies of the data on age (in years) and
sex from the residents of a retirement home.
Age (yrs)
60-69
70-79
Over 79
Total
Male
0.19
0.1
0.11
Female
0.2
0.2
0.2
Total
1
What percentage of residents are females in the age group 70-79?
11) Most patients who undergo surgery make routine recoveries and are discharged as planned, but some patients
experience complications and their discharge is delayed. Jamestown has a large hospital and a small hospital,
each performing major and minor surgeries. Data is collected at each hospital to see how many surgical patients
have their discharges delayed by postsurgical complications. The results are shown in the following table.
Discharge Delayed
Major surgery
Minor surgery
Large hospital
70 of 700
12 of 200
Small hospital
15 of 60
24 of 300
Overall, for what percent of surgical patients was discharge delayed?
12) According to the article "Motion Sickness in Public Road Transport: The Effect of Driver, Route and Vehicle",
seat position within a bus may have some effect on whether one experiences motion sickness. The table below
classifies each person in a random sample of bus riders by the location of his or her seat and whether nausea
was reported.
Front
Middle
Rear
Nausea
58
166
193
No Nausea
870
1163
806
Source: Ergonomics (1999): 1646-1664.
a.
b.
c.
What is the response variable, and what is the explanatory variable?
How do the proportions experiencing nausea compare for the 3 seat positions?
What proportion of all sampled bus riders experienced nausea?
2
13) The article "The Association Between Smoking and Unhealthy Behaviors Among a National Sample of
Mexican-American Adolescents" presents data resulting from a random sample of Mexican-American male
adolescents. Each boy, age 10-18, was classified according to smoking status and his response to a question
asking whether he liked to do risky things.
Smoking Status
Smoker
Nonsmoker
45
46
36
153
Likes Risky Things
Doesn't Like Risky Things
Source: Journal of School Health (1998): 376-379
a. How does the proportion of sampled smokers that likes risky things compare to the proportion of sampled
nonsmokers that likes risky things?
b. What proportion of all sampled Mexican-American male adolescents likes risky things?
14) According to the article "Motion Sickness in Public Road Transport: The Effect of Driver, Route and Vehicle",
seat position within a bus may have some effect on whether one experiences motion sickness. The table below
classifies each person in a random sample of bus riders by the location of his or her seat and whether nausea
was reported.
Front
Middle
Rear
Nausea
58
166
193
No Nausea
870
1163
806
Source: Ergonomics (1999): 1646-1664.
a.
b.
c.
Create a side-by-side bar graph that compares the three seat positions with respect to nausea status.
Summarize the results of the side-by-side bar graph.
Is there an association between motion sickness and seat position in a bus? Explain.
15) The article "The Association Between Smoking and Unhealthy Behaviors Among a National Sample of
Mexican-American Adolescents" presents data resulting from a random sample of Mexican-American male
adolescents. Each boy, age 10-18, was classified according to smoking status and his response to a question
asking whether he liked to do risky things.
Smoking Status
Smoker
Nonsmoker
45
46
36
153
Likes Risky Things
Doesn't Like Risky Things
Source: Journal of School Health (1998): 376-379
a. Create a side-by-side bar graph that compares smoking status with respect to risky things status.
b. Summarize the results of the side-by-side bar graph.
c. Describe how the graph would look if there was not an association between smoking and unhealthy
behaviors among Mexican-American male adolescents.
3
Complete the contingency table and use it to solve the problem.
16) The partially filled contingency table gives the frequencies of the data on age (in years) and sex from the
residents of a retirement home.
Age (yrs)
60-69
70-79
Over 79
Total
Male
11
9
5
Female
9
2
4
Total
What is the proportion of male residents in the age group 60-69?
17) The partially filled contingency table gives the frequencies of the data on age (in years) and sex from the
residents of a retirement home.
Age (yrs)
60-69
70-79
Over 79
Total
Male
15
1
5
Female
5
10
4
Total
What proportion of the female residents are younger than 80?
18) Just how accurate are the weather forecasts we hear every day? The table below compares the daily forecast
with a city's actual weather for a year.
Actual Weather
Forecast Rain
No rain
Rain
33
3
No rain
55
274
GIven that rain was forecast, what proportion of the time did it actually rain?
19) Just how accurate are the weather forecasts we hear every day? The table below compares the daily forecast
with a city's actual weather for a year.
Actual Weather
Forecast Rain
No rain
Rain
25
10
No rain
60
270
Given that rain was forecast, what proportion of the time was there no rain?
4
20) A survey of autos parked in student and staff lots at a large university classified the brands by country of origin,
as seen in the table.
Driver
Brand American
European
Asian
Student
106
30
66
Staff
99
25
25
Given that the driver is a student, what proportion of the cars are Asian?
21) A survey of autos parked in student and staff lots at a large university classified the brands by country of origin,
as seen in the table.
Driver
American
European
Asian
Student
108
36
59
Staff
92
18
56
Given that the car is European, what proportion of the drivers are staff?
Fill in the blank.
22) A ____________________ is a display for two categorical variables.
23) Which of the following displays can be used to describe two categorical variables?
A) stem-and-leaf plot
B) side-by-side box plot
C) histogram
D) contingency table
E) side-by-side dot plot
24) The relationship between the number of games won by a minor league baseball team and the average
attendance at their home games is analyzed. A regression to predict the average attendance from the number of
games won has an r = 0.73. Interpret this statistic.
25) Using advertised prices for used Ford Escorts a linear model for the relationship between a car's age and its
price is found. The regression has an R2 = 87.1%. Describe the relationship
26) A random sample of records of electricity usage of homes gives the amount of electricity used in July and size
(in square feet) of 135 homes. A regression was done to predict the amount of electricity used (in
kilowatt-hours) from size. The residuals plot indicated that a linear model is appropriate. Do you think the
slope is positive or negative? Why?
5
Fill in the blank.
27) A ____________________ association between two quantitative variables is present if y tends to go down as x
goes up.
28) In a positive association between two quantitative variables,
A) y tends to go down as x goes down.
B) the movement of x does not affect the movement of y.
C) y tends to go up as x goes down.
D) y tends to go down as x goes up.
E) none of the above.
29) In a negative association between two quantitative variables,
A) y tends to go up as x goes up.
B) y tends to go down as x goes down.
C) y tends to go up as x goes down.
D) the movement of x does not affect the movement of y.
Determine whether the scatterplot shows little or no association, a negative association, a linear association, a moderately
strong association, or a very strong association (multiple associations are possible).
30)
6
31)
32)
33)
7
34)
35)
Determine which plot shows the strongest linear correlation.
36)
Construct a scatterplot for the data.
37) Below are the Olympic gold medal performances in the men's high jump from 1960 to 1984.
Year High Jump (in.)
1960
85.25
1964
85.75
1968
88.25
1972
87.75
1976
88.50
1980
92.50
1984
92.25
8
Solve the problem.
38) Use a scatter plot to display the data below. All measurements are in milligrams per cigarette.
Brand
Benson & Hedges
Lucky Strike
Marlboro
Viceroy
True
Tar Nicotine
16
1.2
13
1.1
16
1.2
18
1.4
6
0.6
39) The data below represent the numbers of absences and the final grades of 15 randomly selected students from a
statistics class. Use a scatter plot to display the data. Is there a relationship between the students' absences and
their final grades?
Student
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Number of Absences
5
6
2
12
9
5
8
15
0
1
9
3
10
3
11
Final Grade as a Percent
79
78
86
56
75
90
78
48
92
78
81
86
75
89
65
Fill in the blank.
40) A graphical display for two quantitative variables is called a ____________________.
41) Which of the following displays can be used to describe two quantitative variables?
A) all of the above
B) contingency table
C) scatterplot
D) histogram
E) side-by-side bar graph
9
42) Coal is still used as an energy source at The University of Georgia. The processing of raw coal
involves"washing," in which coal ash is removed. Data (see table below) relating the percentage of ash to the
density of a coal particle are given in the article "Quantifying Sampling Precision for Coal Ash Using Gy's
Discrete Model of the Fundamental Error."
Density (g/cc)
% Ash
1.25
1.93
1.325
4.63
1.375
8.95
1.45
15.05
1.55
23.31
Source: Journal of Coal Quality, 2989, 33-39.
Construct a scatterplot to show how the percentage of ash depends on the density of a coal particle. Is a linear
relationship evident from the graph? Explain.
43) Soda is often considered unhealthy because its content is high in both caffeine and refined sugar. But are the
two related? Caffeine and refined sugar contents (in milligrams) of 12 ounces of several brands of soda are
shown on the following scatterplot. The correlation between caffeine and refined sugar is 0.187. Describe the
association.
44) Almost all of the acidity of soda pop comes from the phosphoric acid which is added to give them a sharper
flavor. Is there an association between the pH of the soda and the amount of phosphoric acid (in grams)? The
correlation between pH and phosphoric acid is -0.991. Describe the association.
45) A Science instructor assigns a group of students to investigate the relationship between their grade point
average and their consistency in studying. The correlation between GPA and studying consistency is 0.853.
Assume that the association is linear. Describe the association.
46) Given the length of a human's femur, x, and the length of a human's humerus, y, would you expect a positive
correlation, a negative correlation, or no correlation?
47) Given the supply of a commodity, x, and the price of a commodity, y, would you expect a positive correlation, a
negative correlation, or no correlation?
48) Given the size of a human's brain, x, and their score on an IQ test, y, would you expect a positive correlation, a
negative correlation, or no correlation?
10
49) A random sample of 150 yachts sold in the United States last year was taken. A regression to predict the price
(in thousands of dollars) from length (in feet) has an R2 = 18.9%. What is the correlation between length and
price?
Fill in the blank.
50) The ____________________ is a summary measure that describes the strength of the linear association between
two quantitative variables.
51) The strength of the linear relationship between two quantitative variables may be measured by the
A) y-intercept.
B) scatterplot.
C) residual.
D) slope.
E) correlation.
52) The correlation
A) does not depend on the units of measurement of y or x.
B) depends on the units of measurement of y.
C) depends on the units of measurement of y and x.
D) depends on the units of measurement of x.
E) None of the above.
53) If the correlation is approximately zero, then one can conclude
A) that there is a linear relationship between x and y.
B) that there is no relationship between x and y.
C) that there is a relationship between x and y.
D) that there is no linear relationship between x and y.
54) Which of the following is not a property of r?
A) r is always between -1 and 1.
B) r measures the strength of any kind of relationship between x and y.
C) r does not depend on the units of y or x.
D) The closer r is to zero, the weaker the linear relationship between x and y.
E) r does not depend on which variable is treated as the response variable.
Find the best predicted value of y corresponding to the given value of x.
^
55) Four pairs of data yield r = 0.942 and the regression equation y = 3x. Also, y = 12.75. What is the best predicted
value of y for x = 2.9?
^
56) Eight pairs of data yield r = 0.708 and the regression equation y = 55.8 + 2.79x. Also, y = 71.125. What is the best
predicted value of y for x = 4.2?
57) The regression equation relating dexterity scores (x) and productivity scores (y) for the employees of a company
^
is y = 5.50 + 1.91x. Ten pairs of data were used to obtain the equation. The same data yield r = 0.986 and
y = 56.3. What is the best predicted productivity score for a person whose dexterity score is 20?
11
58) The regression equation relating attitude rating (x) and job performance rating (y) for the employees of a
^
company is y = 11.7 + 1.02x. Ten pairs of data were used to obtain the equation. The same data yield r = 0.863
and y = 80.1. What is the best predicted job performance rating for a person whose attitude rating is 80?
^
59) Nine pairs of data yield r = 0.867 and the regression equation y = 19.4 + 0.93x. Also, y = 64.7. What is the best
predicted value of y for x = 40?
60) The y-intercept is the
A) change in the predicted value of y per unit increase in x.
B) predicted value of y.
C) predicted value of y when x = 0.
D) point where the regression line crosses the x-axis.
E) smallest value for the residual sum of squares.
61) The slope is the
A) predicted value of y when x = 0.
B) point where the regression line crosses the y-axis.
C) predicted value of y.
D) change in the predicted value of y per unit increase in x.
E) smallest value for the residual sum of squares.
Solve the problem.
62) Which statement is true about residuals?
63) A regression line for predicting the selling prices of homes in Chicago is
^
y = 168 + 102x, where x is the square footage of the house. A house with 1500 square feet recently sold for
$140,000. What is the residual for this observation?
64) A regression line for predicting Interenet usage (%) for 39 countries is
^
^
y = -3.61 + 1.55x, where x is the GDP, in thousands of dollars, per capita and y is Internet usage. What is the
residual for one of the 39 countries with GDP per capita of $ 28,000 and actual Internet use of 38 percent?
65) A regression line for predicting Interenet usage (%) for 39 countries is
^
^
y = -3.61 + 1.55x, where x is the GDP, in thousands of dollars, per capita and y is Internet usage. Interpret the
residual for one of the 39 countries with GDP per capita of $ 15,000 and actual Internet use of 20 percent.
66) A regression line for predicting the selling prices of homes in Chicago is
^
y = 168 + 102x, where x is the square footage of the house. Interpret the residual for a house with 1800 square
feet that recently sold for $200,000.
12
67) A random sample of records of electricity usage of homes in the month of July gives the amount of electricity
used and size (in square feet) of 135 homes. A regression was done to predict the amount of electricity used (in
^
kilowatt-hours) from size. Assume that a linear model is appropriate. The model is usage = 1229 + 0.02 size.
What would a negative residual mean for people living in a house that is 2290 square feet?
68) A random sample of records of electricity usage of homes gives the amount of electricity used and size (in
square feet) of 135 homes. A regression to predict the amount of electricity used (in kilowatt-hours) from size
has an r2 =0.71. Assume that a linear model is appropriate. Write a sentence summarizing what r2 says about
this regression.
games won has an r2 = 0.255. The residuals plot indicated that a linear model is appropriate. Write a sentence
summarizing what r2 says about this regression.
games won has an r2 = 0.256. Assume that a linear model is appropriate. What is the correlation between the
average attendance and the number of games won?
(in square feet) of 135 homes. A regression to predict the amount of electricity used (in kilowatt-hours) from
size was completed. Assume that a linear model is appropriate. What are the variables and units in this
regression?
(in square feet) of 135 homes. A regression was done to predict the amount of electricity used (in
kilowatt-hours) from size. Assume that a linear model is appropriate. What units does the slope have?
73) Among the possible lines that can go through data points in a scatterplot, the regression line results from the
least squares method and has the smallest value for the ____________________.
A) correlation
B) intercept
C) slope
D) residual sum
E) residual sum of squares
74) The prediction error for an observation, which is the difference between the actual value and the predicted
value of the response variable, is called ____________________.
A) a residual
B) an outlier
C) a correlation
D) an extrapolation
E) an intercept
13
Fill in the missing information.
_
_
^
75) x sx y sy r y = a + bx
40 20
76)
77)
^
8
11 8 y = ?
sx
_
y
sy r y = a + bx
2.6 1
?
110 ? y = -110 + 40x
_
x
sx
_
y
sy
?
?
18
_
x
^
^
r
^
y = a + bx
^
4 -0.5 y = 30 - 4x
Use a graphing calculator to find the regression line from the given data .
78) x 2 4 5 6
y 7 11 13 20
79) x 0 3 4 5 12
y 8 2 6 9 12
80) x 1.2 1.4 1.6 1.8 2.0
y 54 53 55 54 56
81) Ten students in a graduate program were randomly selected. Their grade point averages (GPAs) when they
entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on
entering the program versus their current GPAs.
x (entering GPA)
3.5
3.8
3.6
3.6
3.5
3.9
4.0
3.9
3.5
3.7
y (current GPA)
3.6
3.7
3.9
3.6
3.9
3.8
3.7
3.9
3.8
4.0
82) Two different tests are designed to measure employee productivity and dexterity. Several employees are
randomly selected and tested with these results.
y (productivity) 23 25 28 21 21 25 26 30 34 36
x (dexterity)
49 53 59 42 47 53 55 63 67 75
14
83) Managers rate employees according to job performance and attitude. The results for several randomly selected
employees are given below.
y (performance) 59 63 65 69 58 77 76 69 70 64
x (attitude)
72 67 78 82 75 87 92 83 87 78
84) Coal is still used as an energy source at The University of Georgia. The processing of raw coal
involves"washing," in which coal ash is removed. Data (see table below) relating the percentage of ash to the
density of a coal particle are given in the article "Quantifying Sampling Precision for Coal Ash Using Gy's
Discrete Model of the Fundamental Error."
Density (g/cc)
% Ash
1.25
1.93
1.325
4.63
1.375
8.95
1.45
15.05
1.55
23.31
Source: Journal of Coal Quality, 2989, 33-39.
Find the regression equation. What is the predicted percentage of ash for a coal particle with density 1.35 g/cc?
Fill in the blank.
85) ____________________ refers to using a regression line to predict y-values for x-values outside the observed
range of data.
Extrapolate from the linear model
86) The figure below shows the life expectancy for persons living in the United States.
The regression analysis of the data yields the following values:
Lifeexp = -353.87 + 0.2157 Year
r2 = 0.9539
Use the regression model to predict the life expectancy in 2010.
15
87) The table below shows the gestation (in days) and average longevity (in years) for a number of different
mammals:
Black Bear
Cat (domestic)
Monkey (Rhesus)
Lion
Horse
Gorilla
Gray Squirrel
Gestation (days)
219
63
166
100
330
258
44
Average Longevity (years)
18
12
15
15
20
20
10
The scatter plot and regression equation are shown below:
The regression analysis of this data yields the following values:
Long = 9.90 + 0.0345 Gestation
r2 = 0.9048
Use this model to predict the average longevity of an African elephant whose gestation is 660 days.
Explain what is wrong with each interpretation. Assume calculations are done correctly.
88) A psychologist does an experiment to determine whether an outgoing person can be identified by his or her
handwriting. She claims that the correlation of 0.89 shows that there is a strong causal relationship between
personality type and handwriting.
89) What is a factor which can cause a high correlation between two variables to be misleading?
Fill in the blank.
90) Predictions about the future using time series data are called ____________________.
91) When an observation has a large effect on the results of a regression analysis, it is said to be
____________________.
92) Why should one always construct a scatterplot first before finding a correlation or a regression line?
93) For an observation to be influential, what two conditions must hold?
16
94) The correlation and the regression line are
A) resistant.
B) nonresistant.
C) causal.
D) unrelated.
Find the lurking variable.
95) A reporter studied the causes of a fire to a house, and established, thanks to a scatterplot a strong correlation
between the damages (in dollars) and the number of firefighters at the scene. Find a lurking variable, if there is
one.
96) A study shows that the amount of chocolate consumed in Canada and the number of automobile accidents is
positively related. Find a lurking variable, if there is one.
97) A study of consumer behavior finds a positive correlation between sales of ice cream and sales of soda. What
might explain the strong correlation?
Fill in the blank.
98) The fact that the direction of an association between two variables can change after we include a third variable
and analyze the data at separate levels of that variable is known as ____________________.
99) When two explanatory variables are both associated with a response variable but are also associated with each
other, there is said to be ____________________.
100) A(n) ____________________ is a variable, usually unobserved, that influences the association between the
variables of primary interest.
A) response variable
B) predictor variable
C) lurking variable
D) independent variable
E) explanatory variable
101) The fact that the direction of an association between two variables can change after we include a third variable
and analyze the data at separate levels of that variable is known as
A) multiple regression.
B) causation.
C) Simpson's paradox.
D) confounding.
E) correlation
17
102) What is the difference between a confounding variable and a lurking variable?
103) Which of the following statements is true?
A) Association does not imply causation.
B) Correlation does not imply causation.
C) The direction of the association between two variables can change with the inclusion of a third variable.
D) None of the above.
E) All of the above.
104) Describe what scatterplots are, and discuss the importance of creating scatterplots.
Find the specified probability distribution of the binomial random variable.
105) A multiple choice test consists of four questions. Each question has five possible answers of which only one is
correct. A student guesses on every question. Find the probability distribution of X, the number of questions she
answers correctly.
106) In one city, 21% of the population is under 25 years of age. Three people are selected at random from the city.
Find the probability distribution of X, the number among the three that are under 25 years of age.
107) In one city, the probability that a person will pass his or her driving test on the first attempt is 0.66. Four people
are selected at random from among those taking their driving test for the first time. Determine the probability
distribution of X, the number among the four who pass the test.
Determine whether a probability model based on Bernoulli trials can be used to investigate the situation. If not, explain.
108) We draw a card from a deck 6 times (replacing the card after each draw) and get 3 kings. How likely is this?
109) We draw a card from a deck 40 times to find the distribution of the suits. After each draw the card is replaced.
110) A pool of possible jurors consists of 15 men and 18 women. A jury of 12 is picked at random from this group.
What is the probability that the jury contains all women?
111) A company realizes that 5% of its pens are defective. In a package of 30 pens, is it likely that more than 6 are
defective? Assume that pens in a package are independent of each other.
112) A study found that 56% of people working for large companies are dissatisfied with their job. What is the
probability that half of the employees at your company are dissatisfied with their job?
18
113) Which of the following is not true about the binomial distribution?
A) The probability of success plus the probability of failure is one.
B) The random variable X is continuous.
C) The probability of success is constant from trial to trial.
D) Each outcome is independent of the other.
114) What are the conditions for a binomial distribution?
115) When is it appropriate to use the binomial distribution to approximate the number of successes in a random
sample of n separate subjects from a population?
116) When is the binomial distribution approximately normal?
117) A college basketball player has probability 0.50 of making any given shot. Her successive shots are
independent. In overtime of an important game, she misses all three shots that she takes. Explain why this does
not mean that she "choked," because it could happen by chance with reasonable probability.
118) Residents of a particular city worry that a waste dump next to the city's water supply may elevate the chance of
cancer for its residents. Of the past 80 deaths of residents, 50 were due to cancer. Investigate whether this
number is unusually high or if it could it be explained by random variation, if actually the probability of a death
being due to cancer was 0.22 in this town. Answer by finding the mean and standard deviation of the
probability distribution of the number of deaths due to cancer and the z-score of the observed result if actually
the probability equals 0.22. Explain your reasoning, stating all assumptions.
Find the mean of the binomial random variable.
119) According to a college survey, 22% of all students work full time. Find the mean for the random variable X, the
number of students who work full time in samples of size 16.
120) A die is rolled 10 times and the number of times that two shows on the upper face is counted. If this experiment
is repeated many times, find the mean for the random variable X, the number of twos.
121) On a 10-question multiple choice test , each question has four possible answers, one of which is correct. For
students who guess at all answers, find the mean for the random variable X, the number of correct answers.
122) The probability that a person has immunity to a particular disease is 0.6. Find the mean for the random variable
X, the number who have immunity in samples of size 26.
123) The probability is 0.7 that a person shopping at a certain store will spend less than $20. For random samples of
28 customers, find the mean number who spend less than $20.
Find the standard deviation of the binomial random variable.
124) According to a college survey, 22% of all students work full time. Find the standard deviation for the random
variable X, the number of students who work full time in samples of size 16.
19
125) A die is rolled 18 times and the number of twos that come up is tallied. If this experiment is repeated many
times, find the standard deviation for the random variable X, the number of twos.
126) On a multiple choice test with 16 questions, each question has four possible answers, one of which is correct. For
students who guess at all answers, find the standard deviation for the random variable X, the number of correct
answers.
127) The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the
standard deviation for the random variable X, the number of seeds germinating in each batch.
128) The probability of winning a certain lottery is 1/51,949. For people who play 560 times, find the standard
deviation for the random variable X, the number of wins.
Find the indicated probability.
129) An archer is able to hit the bull's-eye 55% of the time. If she shoots 8 arrows, what is the probability that she
gets exactly 4 bull's-eyes? Assume each shot is independent of the others.
130) A tennis player makes a successful first serve 59% of the time. If she serves 7 times, what is the probability that
she gets exactly3 first serves in? Assume that each serve is independent of the others.
131) A multiple choice test has 10 questions each of which has 4 possible answers, only one of which is correct. If
Judy, who forgot to study for the test, guesses on all questions, what is the probability that she will answer
exactly 3 questions correctly?
132) In one city, the probability that a person will pass his or her driving test on the first attempt is 0.68. 11 people
are selected at random from among those taking their driving test for the first time. What is the probability that
among these 11 people, the number passing the test is between 2 and 4 inclusive?
133) Suppose that11% of people are left handed. If 6 people are selected at random, what is the probability that
exactly 2 of them are left handed?
134) A basketball player has made 70% of his foul shots during the season. If he shoots 3 foul shots in tonight's game,
what is the probability that he makes all of the shots?
135) Police estimate that 25% of drivers drive without their seat belts. If they stop 6 drivers at random, find the
probability that all of them are wearing their seat belts.
136) The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest
that 20% of people with home-based computers have access to on-line services. Suppose that 15 people with
home-based computers were randomly and independently sampled. What is the probability that exactly 5 of
those sampled have access to on-line services at home?
137) The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest
that 20% of people with home-based computers have access to on-line services. Suppose that 15 people with
home-based computers were randomly and independently sampled. What is the probability that at least 1 of
those sampled have access to on-line services at home?
20
Obtain the probability distribution of the random variable.
138) When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below:
HHHH HHHT HHTH HHTT
HTHH HTHT HTTH HTTT
THHH THHT THTH THTT
TTHH TTHT TTTH TTTT
Let X denote the total number of tails obtained in the four tosses. Find the probability distribution of the random
variable X. Leave your probabilities in fraction form.
139) When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below.
(1, 1)
(2, 1)
(3, 1)
(4, 1)
(5, 1)
(6, 1)
(1, 2) (1, 3) (1, 4)
(2, 2) (2, 3) (2, 4)
(3, 2) (3, 3) (3, 4)
(4, 2) (4, 3) (4, 4)
(5, 2) (5, 3) (5, 4)
(6, 2) (6, 3) (6, 4)
(1, 5) (1, 6)
(2, 5) (2, 6)
(3, 5) (3, 6)
(4, 5) (4, 6)
(5, 5) (5, 6)
(6, 5) (6, 6)
Let X denote the absolute value of the difference of the two numbers. Find the probability distribution of X. Give
the probabilities as decimals rounded to three decimal places.
140) When two balanced dice are rolled, 36 equally likely outcomes are possible as shown below.
(1, 1)
(2, 1)
(3, 1)
(4, 1)
(5, 1)
(6, 1)
(1, 2) (1, 3) (1, 4)
(2, 2) (2, 3) (2, 4)
(3, 2) (3, 3) (3, 4)
(4, 2) (4, 3) (4, 4)
(5, 2) (5, 3) (5, 4)
(6, 2) (6, 3) (6, 4)
(1, 5) (1, 6)
(2, 5) (2, 6)
(3, 5) (3, 6)
(4, 5) (4, 6)
(5, 5) (5, 6)
(6, 5) (6, 6)
Let X denote the smaller of the two numbers. If both dice come up the same number, then X equals that common
value. Find the probability distribution of X. Leave your probabilities in fraction form.
141) Which of the following is a property of a discrete probability distribution?
A) Each interval has probability between 0 and 1. This is the area under the curve, above that interval.
B) The sum of the probabilities for all the possible x values equals 1.
C) The interval containing all possible values has probability equal to 1, so the total area under the curve
equals 1.
D) Both a. and b.
21
142) Which of the following is a property of a continuous probability distribution?
A) For each x, the probability p(x) falls between 0 and 1.
B) The sum of the probabilities for all the possible x values equals 1.
C) The interval containing all possible values has probability equal to 1, so the total area under the curve
equals 1.
D) Both a. and b.
Determine the possible values of the random variable.
143) Suppose a coin is tossed four times. Let X denote the total number of tails obtained in the four tosses. What are
the possible values of the random variable X?
144) Suppose that two balanced dice are rolled. Let Y denote the product of the two numbers. What are the possible
values of the random variable Y?
145) Suppose that two balanced dice, a red die and a green die, are rolled. Let Y denote the value of G - R where G
represents the number on the green die and R represents the number on the red die. What are the possible
values of the random variable Y?
146) For a randomly selected student in a particular high school, let Y denote the number of living blood related
grandparents of the student. What are the possible values of the random variable Y?
147) The following table displays a frequency distribution for the number of siblings for students in one middle
school. For a randomly selected student in the school, let X denote the number of siblings of the student. What
are the possible values of the random variable X?
Number of siblings
0
1
2 3 4 5 6 7
Frequency 189 245 102 42 24 13 5 2
Find the mean of the given probability distribution.
148) The random variable X is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate
office. Its probability distribution is given in the table.
x P(X = x)
0
0.24
1
0.01
2
0.12
3
0.16
4
0.01
5
0.14
6
0.11
7
0.21
149) The random variable X is the number of tennis balls ordered by customers at Tennis Online. Its probability
distribution is given in the table.
x
3
6
9
12
15
P(X = x) 0.14 0.22 0.36 0.18 0.10
22
150) The random variable X is the number of people who have an associate's degree in a randomly selected group of
four adults from a particular town. Its probability distribution of X, based on 1000 random samples of size 4
adults, is given in the table.
x P(X = x)
0
0.026
1
0.153
2
0.346
3
0.345
4
0.130
151) The random variable X is the number that shows up when a loaded die is rolled. Its probability distribution is
given in the table.
x P(X = x)
1
0.14
2
0.12
3
0.12
4
0.10
5
0.13
6
0.39
152) The random variable X is the number of siblings of a student selected at random from a particular secondary
school. Its probability distribution is given in the table.
x
0
7
P(X = x)
24
1
13
48
2
3
16
3
7
48
4
1
16
5
1
24
153) The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 0.6275, 0.3102,
0.0575, 0.0047 and 0.0001, respectively. Round answer to the nearest hundredth.
154) A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day
are 0.50, 0.40, 0.09, and 0.01 respectively.
155) Computer chips often contain surface imperfections. For a certain type of computer chip, 9% contain no
imperfections, 22% contain 1 imperfection, 26% contain 2 imperfections, 20% contain 3 imperfections, 12%
contain 4 imperfections, and the remaining 11% contain 5 imperfections. Let X represent the number of
imperfections in a randomly chosen chip.
a.
b.
Specify the probability distribution for the number of imperfections in a randomly chosen chip.
Find the expected number of imperfections in a randomly chosen chip
156) Let X = the number of days in the past week in which a randomly selected person felt anxious and tense.
According to a recent General Social Survey, the probabilities for the potential values of X for adult Americans
are approximately:
x
0
1
P(x) 0.28 0.20
a.
b.
2
3
4
5
6
0.15 0.11 0.07 0.07 0.02
Does this refer to a discrete or a continuous random variable?
Find the mean of this probability distribution. Interpret.
23
7
0.10
Use the empirical rule to solve the problem.
157) The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a
standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure that
lies within 3 standard deviations of the mean?
158) The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a
standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure
between 96 mmHg and 144 mmHg?
159) The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 360 hours and a
standard deviation of 5 hours. What percentage of the bulbs have lifetimes that lie within 1 standard deviation
of the mean?
160) The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 360 hours and a
standard deviation of 8 hours. What percentage of the bulbs have lifetimes that lie within 2 standard deviations
of the mean?
161) At one college, GPA's are normally distributed with a mean of 2.6 and a standard deviation of 0.4. What
percentage of students at the college have a GPA between 2.2 and 3?
162) The amount of Jennifer's monthly phone bill is normally distributed with a mean of $79 and a standard
deviation of 9. Fill in the blanks.
68% of her phone bills are between $___ and $___.
163) The annual precipitation for one city is normally distributed with a mean of 390 inches and a standard
deviation of 3.2 inches. Fill in the blanks.
In 95% of the years, the precipitation in this city is between ___ and ___ inches.
164) Which of the following is not true about the standard normal distribution?
A) The area under the standard normal curve to the left of z = 0 is negative.
B) About 68% of its observations fall between -1 and 1.
C) The standard normal curve is symmetric about 0.
D) About 95% of its observations fall between -2 and 2.
E) The total area under the standard normal curve is 1.
165) Serum cholesterol is an important risk factor for coronary disease. The level of serum cholesterol is
approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. What is
the interquartile range of the level of serum cholesterol?
Use a table of areas to find the specified area under the standard normal curve.
166) The area that lies between 0 and 3.01
24
167) The area that lies to the left of 1.13
168) The area that lies between -1.10 and -0.36
169) The area that lies to the right of -1.82
170) The area that lies to the right of 0.59
171) The shaded area shown
Use a table of areas for the standard normal curve to find the required z-score.
172) Find the z-score for which the area under the standard normal curve to its left is 0.96
173) Find the z-score having area 0.09 to its left under the standard normal curve.
174) Find the z-score for which the area under the standard normal curve to its left is 0.04
175) Find the z-score having area 0.86 to its right under the standard normal curve.
176) Suppose that property taxes on homes in Columbus, Ohio, are approximately normal in distribution, with a
mean of $3000 and a standard deviation of $1000. The property tax for one particular home is $3500.
a.
b.
Find the z-score for that property tax value.
What proportion of the property taxes exceeds $3500?
approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. If the
clinically desirable range for serum cholesterol is < 200 mg/dL, what is the probability that a randomly selected
person will have a clinically desirable level of serum cholesterol?
approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. If serum
cholesterol levels of over 250 mg/dL indicate a high-enough risk for heart disease to warrant treatment, what is
the probability that a randomly selected person will need treatment?
approximately normally distributed with a mean of 219 mg/dL and a standard deviation of 50 mg/dL. If the
clinically desirable range for serum cholesterol is < 200 mg/dL and serum cholesterol levels of over 250 mg/dL
indicate a high-enough risk for heart disease to warrant treatment, what is the probability that a randomly
selected person will have a borderline high serum cholesterol level (that is, > 200, but < 250 mg/dL)?
25
Find the indicated probability for the normally distributed variable.
180) The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a
standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches?
181) The incomes of trainees at a local mill are normally distributed with a mean of $1,100 and a standard deviation
$150. What percentage of trainees earn less than $900 a month?
182) The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard
deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected bottle will be less
than 32 oz?
183) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a
standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 200
and 275.
184) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a
standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 170
and 220.
185) The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation
of 15 days. What is the probability that a pregnancy lasts at least 300 days?
186) Assume that the weights of quarters are normally distributed with a mean of 5.67 g and a standard deviation
0.070 g. A vending machine will only accept coins weighing between 5.48 g and 5.82 g. What percentage of legal
quarters will be rejected?
187) The diameters of pencils produced by a certain machine are normally distributed with a mean of 0.30 inches
and a standard deviation of 0.01 inches. What is the probability that the diameter of a randomly selected pencil
will be less than 0.285 inches?
188) The weekly salaries of teachers in one state are normally distributed with a mean of $490 and a standard
deviation of $45. What is the probability that a randomly selected teacher earns more than $525 a week?
189) In one region, the September energy consumption levels for single-family homes are found to be normally
distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. For a randomly selected home, find
the probability that the September energy consumption level is between 1100 kWh and 1225 kWh.
Find the probability involving proportion.
190) Assume that 20% of students at a university wear contact lenses. We randomly pick 200 students. What is the
probability that more than 22% of this sample wear contact lenses?
191) A candy company claims that its jelly bean mix contains 15% blue jelly beans. Suppose that the candies are
packaged at random in small bags containing about 200 jelly beans. What is the probability that a bag will
contain more than 20% blue jelly beans?
192) When a truckload of peaches arrives at a packing plant, a random sample of 125 is selected and examined. The
whole truckload will be rejected if more than 8% of the sample is unsatisfactory. Suppose that in fact 11% of the
oranges on the truck do not meet the desired standard. What's the probability that the shipment will be
accepted anyway?
26
193) Researchers believe that 6% of children have a gene that may be linked to a certain childhood disease. In an
effort to track 50 of these children, researchers test 950 newborns for the presence of this gene. What is the
probability that they find enough subjects for their study?
194) Researchers believe that 7% of children have a gene that may be linked to a certain childhood disease. In an
effort to track 50 of these children, researchers test 950 newborns for the presence of this gene. What is the
probability that they do not find enough subjects for their study?
195) The closer p is to 0 or 1, the larger n must be in order for the sampling distribution of the sample proportion to
A) be approximately binomial.
B) be centered at p.
C) have a standard error equal to [n(p)(1-p)].
D) be approximately normal.
E) have a standard error equal to [p(1-p)/n].
Find the mean/standard error of the sampling distribution of the proportion.
196) Based on past experience, a bank believes that 8% of the people who receive loans will not make payments on
time. The bank has recently approved 600 loans. Describe the sampling distribution model of the proportion of
clients in this group who may not make timely payments.
197) Assume that 26% of students at a university wear contact lenses. We randomly pick 300 students. Describe the
sampling distribution model of the proportion of students in this group who wear contact lenses.
198) A realtor has been told that 43% of homeowners in a city prefer to have a finished basement. She surveys a
group of 300 homeowners randomly chosen from her client list. Describe the sampling distribution model of
the proportion of homeowners in this sample who prefer a finished basement.
199) A candy company claims that its jelly bean mix contains 21% blue jelly beans. Suppose that the candies are
packaged at random in small bags containing about 400 jelly beans. Describe the sampling distribution model
of the proportion of blue jelly beans in a bag.
200) A recent poll of 1000 British adults asked, "If there were a referendum on the issue, would you favor Britain
becoming a republic or remaining a monarchy?" (www.mori.com/polls). Suppose that the population
proportion favoring the monarchy equals 0.70. (This was, in fact, the value for the sample proportion.) For a
random sample of 1000 residents, let X denote the number in this category.
a. Find the mean of the probability distribution of X.
b. Find the standard deviation of the probability distribution of X.
c. What range of values falls within 3 standard deviations of the mean?
Explain why it is unlikely that X will fall outside this interval.
27
201) Eurobarometer recently took a poll to gauge the approval of adults for the European currency. Of the 1000
people sampled in a particular country in the European Union, consider the sample proportion of people who
indicate approval of the euro.
a. Find the mean and standard error of the sampling distribution for this sample proportion, if the population
proportion equals 0.67.
b. What shape would you expect this sampling distribution to have? Explain. Within what limits would the
sample proportion almost certainly fall? Explain.
Solve the problem.
202) In one region, the September energy consumption levels for single-family homes are found to be normally
distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. If 50 different homes are randomly
selected, find the probability that their mean energy consumption level for September is greater than 1075 kWh.
203) Assume that women's heights are normally distributed with a mean of 63.6 inches and a standard deviation of
2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9
inches and 64.0 inches.
204) Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a
standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a
mean replacement time less than 9.1 years.
205) Human body temperatures are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F.
If 19 people are randomly selected, find the probability that their mean body temperature will be less than
98.50°F.
206) For women aged 18-24, systolic blood pressures (in mm Hg) are normally distributed with a mean of 114.8 and
a standard deviation of 13.1. If 23 women aged 18-24 are randomly selected, find the probability that their
mean systolic blood pressure is between 119 and 122.
207) A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier
shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected,
find the probability that their mean rebuild time exceeds 8.7 hours.
208) A population distribution has mean 50 and standard deviation 20. For a random sample of size 100, the
sampling distribution of the sample mean has
A) mean 50/100 and standard error 20/100.
B) mean 50 and standard error 0.20.
C) mean 50/10 and standard error 20/10.
D) mean and standard error that are unknown unless we know the exact shape of the population distribution.
E) mean 50 and standard error 2.
28
209) The sampling distribution of sample means is from a highly skewed population with µ = 4.47 and
repeated random samples of size 100 from this population:
= 1.40. For
a. Find the mean and standard error of the sampling distribution of the sample mean.
b. Explain why the sampling distribution of the sample mean is bell-shaped, even though the population was
highly skewed.
210) To estimate the mean acreage of ranches in Alberta, Canada, a researcher plans to obtain the acreage for a
random sample of 64 farms. Results from an earlier study suggest that 800 acres is a reasonable guess for the
standard deviation of ranch size.
a. Find the probability that the sample mean acreage falls within 100 acres of the population mean acreage.
b. If the researcher can increase n above 64, will the probability that the sample mean falls within 100 acres of
the population mean increase or decrease? Why?
211) .In North America, female adult heights are approximately normal with µ = 65 inches and = 3.5 inches. The
heights of 50 females were measured at a national collegiate volleyball tournament. The sample mean height
was found to be 70 inches.
a. Using the population parameters given above, what is the probability of obtaining a sample mean height of
70 inches or higher with a random sample of n = 50?
b. Does this probability make you question the population mean stated for female heights? Justify why you
believe this sample mean may not be representative of the population of female heights.
For samples of the specified size from the population described, find the mean and standard error if the sampling
distribution of the sample mean x.
212) The mean and the standard deviation of the sampled population are, respectively, 113.9 and 32.1.
n = 64
213) The mean and the standard deviation of the sampled population are, respectively, 77.4and 4.0.
n = 225
214) The National Weather Service keeps records of snowfall in mountain ranges. Records indicate that in a certain
range, the annual snowfall has a mean of 97 inches and a standard deviation of 16 inches. Suppose the snowfalls
are sampled during randomly picked years. For samples of size 64 determine the mean and standard deviation
of x.
215) One barge from Inland Waterways, Inc. can carry a load of 5516.8 lb. Records of past trips show that the weight
of cans that it carries have a mean of 82 lb and standard deviation of 16 lb. For samples of size 64 find the mean
and standard deviation of x.
216) One truck from Pee Dee Trucking, Inc. can carry a load of 4876.8 lb. Records of past trips show that the weights
of boxes that it carries have a mean of 72 lb and a standard deviation of 16 lb. For samples of size 64, find the
mean and standard deviation of x.
29
217) The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 25
days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 266
days and 268days.
218) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard
deviation of 2.5 inches. If 100 women are randomly selected, find the probability that they have a mean height
greater than 63.0 inches.
219) Assume that the heights of women are normally distributed with a mean of 63.6 inches and a standard
deviation of 2.5 inches. If 75 women are randomly selected, find the probability that they have a mean height
between 63 and65 inches.
220) Assume that the heights of men are normally distributed with a mean of 67.9inches and a standard deviation of
2.8 inches. If 64 men are randomly selected, find the probability that they have a mean height greater than 68.9
inches.
221) The body temperatures of adults are normally distributed with a mean of 98.6° F and a standard deviation of
0.60° F. If 36 adults are randomly selected, find the probability that their mean body temperature is greater than
98.4° F.
222) Assume that the salaries of elementary school teachers in the United States are normally distributed with a
mean of $32,000 and a standard deviation of $3000. If 100 teachers are randomly selected, find the probability
that their mean salary is greater than $32,500.
223) Assume that blood pressure readings are normally distributed with a mean of 120 and a standard deviation of
8. If 100 people are randomly selected, find the probability that their mean blood pressure will be greater than
122.
224) The average number of pounds of red meat a person consumes each year is 196 with a standard deviation of 22
pounds (Source: American Dietetic Association). If a sample of 50 individuals is randomly selected, find the
probability that the mean of the sample will be less than 200 pounds.
225) A coffee machine dispenses normally distributed amounts of coffee with a mean of 12 ounces and a standard
deviation of 0.2 ounce. If a sample of 9 cups is selected, find the probability that the mean of the sample will be
greater than 12.1 ounces.
226) The ____________________ is considered powerful in statistics because it works for any population distribution
provided the sample size is sufficiently large and the population mean and standard deviation are known.
A) binomial distribution
B) central limit theorem
C) law of large numbers
D) normal distribution
E) empirical rule
30

Chapters 3 and 6 Sample Problems Provide an appropriate response.

Transcription

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