Math 116 - Final Review

Math 116 - Final Review
Name___________________________________
You may assume that the samples are independent and that the variable under consideration is normally distributed on
both populations. The two population standard deviations cannot be assumed to be equal.
1) A paint manufacturer wishes to compare the drying times of two different types of paint. Independent random
samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces. The drying
times were recorded. The summary statistics are as follows.
Type A
Type B
x 1 = 76.0 x 2 = 64.0
s = 4.5
s = 5.1
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2
n = 11
n =9
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2
a) Do the data suggest that Type A paint take longer to dry? Test the claim at the 0.01 level of significance.
b) Will the conclusion be different if you use a different significance level?
c) Determine a 98% confidence interval for the difference, µ - µ , between the mean drying time for paint of
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types A and B.
d) What is the interval suggesting about µ and µ ? Are they equal or is one of them larger than the other?
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2) 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? Show procedure.
You may assume that the samples are independent and that the variable under consideration is normally distributed on
both populations. The two population standard deviations cannot be assumed to be equal.
3) A researcher was interested in comparing the amount of time spent watching television by women and by men.
Independent random samples of 14 women and 17 men were selected and each person was asked how many
hours he or she had watched television during the previous week. The summary statistics are as follows.
Sample 1 (women) Sample 2 (men)
x 1 = 12.6
s = 3.9
1
n = 14
1
x 2 = 13.6
s = 5.2
2
n = 17
2
a) Do the data provide sufficient evidence to conclude that the on average, women watch less hours of TV per
week than men? Test at the 5% significance level.
b) Construct a 90% confidence interval estimate for the difference µ - µ
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2
c) What is the interval suggesting about µ and µ ?
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2
4) The monthly expenditures on food by single adults living in one neighborhood of Los Angeles are normally
distributed with a mean of $370 and a standard deviation of $80.
a) Describe the distribution of sample means for sampels of size 16. Give the shape, mean and standard error.
b) Determine the probability that a sample of size 16 thas a mean expenditure within $36 of the population
mean expenditure of $370.
5) Assume that women have heights that are normally distributed with a mean of 63.6 inches and a standard
deviation of 2.5 inches. Find the value of the quartile Q3 .
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6) A survey of 865 voters in one state reveals that 459 favor the death penalty.
a) At the 1% significance level can we support the claim that the majority of voters in the state favor the death
penalty?
b) Construct the 98% confidence interval for the true proportion of all voters in the state who favor the death
penalty. Does the interval support the claim stated in (a)? Explain.
7) The heights of people in a certain population are normally distributed with a mean of 65 inches and a standard
deviation of 3.4 inches. Determine the sampling distribution of the mean for samples of size 42.
A) Normal, mean = 65 inches, standard deviation = 0.52 inches
B) Normal, mean = 65 inches, standard deviation = 0.08 inches
C) Approximately normal, mean = 65 inches, standard deviation = 0.08 inches
D) Normal, mean = 65 inches, standard deviation = 3.4 inches
Assume that the differences are normally distributed.
8) A coach uses a new technique in training middle distance runners. The times for 8 different athletes to run 800
meters before and after this training are shown below.
Athlete
A
B
C
D
E
F
G
H
Time before
training (seconds) 116.9 108.4 114.1 112.8 116.9 117.1 114.8 111.5
Time after training
(seconds) 117.5 107.1 111.7 113.6 115.1 117.2 111.2 107.6
a) Do the data suggest that the training helps to improve the athletes' times for the 800 meters? Perform a paired
t-test at the 5% significance level.
b) Construce a 90% confidence interval estimate for the mean of the differences.
c) What does the interval suggest?
9) In a random sample of 360 women, 65% favored stricter gun control laws. In a random sample of 220 men, 60%
favored stricter gun control laws.
a) Test the claim that the proportion of women favoring stricter gun control is higher than the proportion of
men favoring stricter gun control. Use a significance level of 0.05.
b) Construct a 90% confidence interval estimate for p1 - p2.
c) What is the interval suggesting? p1 = p2, p1 < p2, p1 > p2?
10) A manufacturer considers his production process to be out of control when defects exceed 3%. In a random
sample of 85 items, the defect rate is 5.9% but the manager claims that this is only a sample fluctuation and
production is not really out of control.
a) At the 0.05 level of significance, do the data provide sufficient evidence that the percentage of defects exceeds
3%?
b) Construct a 90% confidence interval estimate for the proportion of defective parts.
c) What does the interval suggest?
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11) A light-bulb manufacturer advertises that the average life for its light bulbs is 900 hours. A random sample of
15 of its light bulbs resulted in the following lives in hours.
995
590
510
539
739
917
571
555
916
728
664
693
708
887
849
a) Construct a box plot and a normal probability plot to confirm that there are no outliers and that the
population is approximatelly normally distributed.
b) At the 10% significance level, do the data provide evidence that the mean life for the company's light bulbs
differs from the advertised mean?
c) Construct a 90% confidence interval estimate for the life of the light bulbs. What does the interval suggest?
12) The forced vital capacity (FVC) is often used by physicians to assess a person's ability to move air in and out of
their lungs. It is the maximum amount of air that can be exhaled after a deep breath. For adult males, the
average FVC is 5.0 liters. A researcher wants to perform a hypothesis test to determine whether the average
forced vital capacity for women differs from this value. The mean forced vital capacity for a random sample of
85 women was 4.8 liters with a standard deviation of 0.9 liters
a) Do the data provide sufficient evidence to conclude that the mean forced vital capacity for women differs
from the mean value for men of 5.0 liters? Perform the appropriate hypothesis test using a significance level of
0.05.
b) Construct a 95% confidence interval estimate for the mean forced vital capacity of women.
c) What does the interval suggest?
13) Ten different families are tested for the number of gallons of water a day they use before and after viewing a
conservation video. Do the data suggest that the mean amount after the viewing is lower than the mean amount
before the viewing?
a) Perform a paired t-test at the 5% significance level.
b) Construce a 90% confidence interval estimate for the number of gallons of water used per day.
c) What does the interval suggest?
Before 33 33 38 33 35 35 40 40 40 31
After
34 28 25 28 35 33 31 28 35 33
14) 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?
A) 1.62%
B) 1.94%
C) 2.48%
D) 0.0194%
15) A newspaper in a large midwestern city reported that the National Association of Realtors said that the mean
home price last year was $116,800. The city housing department feels that this figure is too low. They randomly
selected 66 home sales and obtained a sample mean price of $118,900. Assume that the population standard
deviation is $3,700.
a) Using a 5% level of significance, perform a hypothesis test to determine whether the population mean is
higher than $116,800.
b) Construct a 90% confidence interval estimate for the mean home price in the midwestern city.
c) What is the confidence interval suggesting about the mean home price of houses in the area?
16) When bears were anesthetized, researchers measured the distance (in inches) around their chest and they
weighed the bears (in pounds). The results are given below for eight male bears.
x Chest (in) 26 45 54 49 41 49 44 19
y Weight (lb) 90 344 416 348 262 360 332 34
a) Determine whether there is a significant linear correlation.
b) If so, find the regresion equation.
c) Find the best predicted weight of a bear with a chest size of 52 inches.
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17) You wish to estimate the proportion of all voters in California who plan to vote in favor of a certain ballot
measure. How large of a sample should we select to be 99% confident that the sample proportion x-bar is
within 0.015 from the population proportion p.
a) Assume that in a preliminary study 29% of the voters sampled voted in favor.
b) Assume we have no results from a preliminary study.
18) Scores on a certain test are normally distributed with a standard deviation of 8.77. A researcher wishes to
estimate the mean score achieved by all adults on the test. Find the sample size needed to assure with 95.44
percent confidence that the sample mean will not differ from the population mean by more than 3 units.
Provide an appropriate response.
19) It is believed that 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
a.
b.
c.
Nausea
58
166
193
No Nausea
870
1163
806
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?
20) Using advertised prices for a certain used car model, 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
A) No association
B) Positive, strong linear relationship. As the age increases the price goes up.
C) Negative, strong linear relationship. As the age increases the price goes down.
D) Negative, weak linear relationship. As the age decreases the price goes down.
E) Positive, weak linear relationship. As the age increases the price goes down.
21) Would you expect the following pair of variables measured for 200 individuals aged 18-32 to have a positive
association, negative association, or no association: amount of time spent exercising per week and height?
A) positive association
B) no association
C) negative association
22) It is believed that 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
a.
b.
c.
Nausea
58
166
193
No Nausea
870
1163
806
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.
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23) The regression equation predicting the average weight of a male aged 18-24 (y) based on his height (x) is given
^
by y = -172.63 + 4.842x. Describe the correlation between height and weight based on the slope of the regression
line.
A) positive correlation since the magnitude of the slope is large relative to the intercept
B) negative correlation since the magnitude of the slope is small relative to the intercept
C) negative correlation since the slope is negative
D) positive correlation since the slope is positive
24) The regression equation predicting the average weight of a male aged 18-24 (y) based on his height (x) is given
^
by y = -172.63 + 4.842x. What is the best prediction for the weight of a male aged 18-24 who is 70 inches tall?
Round your answer to the nearest pound.
25) The regression equation predicting the average weight of a male aged 18-24 (y) based on his height (x) is given
^
by y = -172.63 + 4.842x. Interpret the slope of the regression line.
A) for every unit increase in weight, the predicted height increases by 4.842 pounds
B) for every unit increase in height, the predicted weight increases by 4.842 pounds
C) for every unit increase in height, the predicted weight decreases by 4.842 pounds
D) for every unit increase in weight, the predicted height decreases by 4.842 pounds
_______________________________________________________________________________________________
26) ANSWERS:
1) a) t = 5.517, p = 0.00002; Support the claim that type A paint takes longer to dry
b) No, the p-value is so small that the p-value will always be lower than the significance level
c) (6.38 ,17.62)
d) The interval is completely on the positive side, this means µ is larger than µ , which is the same
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conclusion as in the hypothesis testing
2) 0.4013
3) a) Test statistic: t = -0.611 (not far from zero)
p-value = 0.273 (large!)
Critical value: t = -1.701 (even though we don't need this, practice finding this anyways)
Do not reject H . At the 5% significance level, the data do not provide sufficient evidence to conclude that the
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mean time for women is less than the mean time for men. Notice that the conclusion will be the same for any
significance level because the p value is large.
b) (-3.781 , 1.7807)
c) Since the interval contains zero, it's possible for mu1 to be equal to mu2, which is the same conclusion as the
HT.
4) a) x-bar (n=16) is normal because the population is normal. The mean is 370 and the standard error is 20. b)
92.82%
5) 65.3 inches
6) a) z = -1.8
p = 0.036
Do not reject Ho. At the 1% significance level we do not have enough evidence to support the claim that.....
b) from 0.49116 to .57011
Since the interval contains .50, it may be equal to it. Same conclusion as in part (a).
7) A
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8)a) Test statistic: t = 2.227
p = 0.03
Critical value: t = 1.895
Reject H . At the 5% significance level, the data provide sufficient evidence to conclude that the training helps
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to improve the athletes' times for the 800 meters.
b) (.21441, 2.6606)
c) It suggests that the mean of the differences is greater than zero. Same conclusion as in (a)
9) a) Test statistic: z = 1.21.
p-value = .113
Critical value: z = 1.645. (practice finding this)
Fail to reject the null hypothesis. There is not sufficient evidence to support the claim that the proportion of
women favoring stricter gun control is higher than the proportion of men favoring stricter gun control.
b) (-.0183, .11827)
c) This isterval suggests that p1 and p2 may be equal. Notice that it agrees with the conclusion of the hypothesis
testing.
10) a) Test statistic: z = 1.57.
p = .0596
Critical value: z = 2.33. (practice finding this)
Fail to reject the null hypothesis. There is not sufficient evidence to conclude that the percentage of defects
exceeds 3%.
b) (.0168, .1008)
c) Since the interval contains 0.03, it may be equal to 0.03. Same conclusion as in part (a)
11) a) Test statistic: t = -4.3421.
p = 0.000676
Reject H : There is sufficient evidence to support the claim that the true mean life differs from the advertised
0
mean. (Notice that at any of the common significance levels the conclusion will be the same)
b) 652.7 to 795.43
c) It's completely below 900. Same conclusion as in (a)
12) a) z = -2.05
P-value = 0.0436
Reject H . At the 5% significance level, the data provide sufficient evidence to conclude that the mean forced
0
vital capacity for women differs from 5.0 liters.
b) (4.6059, 4.9941)
c) The interval is completely below 5. It supports the claim that the mean forced vital capacity for women differs
from 5.0 liters.
13) a) Test statistic: t = 2.894
p = 0.009
Critical values: t = ±2.262
Reject H . At the 5% significance level, the data provide sufficient evidence to conclude that the mean amount
0
after the viewing differs from the mean amount before the viewing.
b) 1.7595 to 7.8405
c) It's completely above zero. It suggests that the mean of the differences is larger than zero which supports the
claim that after viewing the video the families consume less water a day.
14) B
15) a) Test statistic: z = 4.61.
p = 0.000002
Critical value: z = 1.645. Reject H o: µ = $116,800.There is sufficient evidence to support the claim that the mean
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is higher than $116,800.
b) (118151, 119649)
c) Since the interval is completely above 116,800, it suggests that mu is higher. Same conclussion as in part (a)
16) a) r = .993 , Significant linear correlation.
b) y = 11.3 x - 187
c) 400.6 lb
17) a) 6068, b) 7368; If there is an estimate from a previous study we prefer to use it because it produces a
smaller sample size, which implies a cheaper study.
18) 35
19) a. Nausea status, whether or not a bus rider reported nausea, is the variable of interest. The seat position
(front, middle, or rear) is the variable that defines 3 groups to be compared on their nausea status. So nausea
status is the response variable and seat position is the explanatory variable; b. The proportion of bus riders
sitting up front who experienced nausea was 0.0625. The proportion of bus riders sitting in the middle who
experienced nausea was 0.1249. The proportion of bus riders sitting at the rear who experienced nausea was
0.1932. The proportion of bus riders sitting in the middle who experienced nausea was almost twice as high as
the proportion of bus riders sitting up front who experienced nausea. The proportion of bus riders sitting at the
rear who experienced nausea was over one and a half times as high as the proportion of bus riders sitting in the
middle who experienced nausea; c. 0.1281.
20) C
21) B
22) a.
b. The graph clearly shows that the proportion of bus riders experiencing nausea is highest for those bus riders
seated at the rear and lowest for those bus riders seated up front. The proportion of bus riders sitting at the rear
who experienced nausea is approximately three times as high as the proportion of bus riders sitting up front
who experienced nausea; c. Yes, because it is evident from the graph that as one's seat position moves away
from the front of the bus the incidence of motion sickness increases.
23) D 24) 166
25) B
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