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Prepared by: Paolo Bautista
Preliminaries
 We wish to test whether a particular assumption/claim
regarding the population is true or not.
 Null Hypothesis (H0) – original assumption
 Alternative Hypothesis (H1)
 Determine a critical value to determine whether or not
to reject Ho
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Errors
 Type I Error – reject H0 when in fact it is true
 Type II Error – fail to reject H0 when it is false
Fail to Reject
Reject
Null Hypothesis is Null Hypothesis is
True
False
Correct Decision
Type II Error (β)
Type I Error (α)
Correct Decision
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Steps in Hypothesis Testing
1.
2.
3.
4.
5.
Write the null and alternative hypotheses.
Indicate the level of significance.
Determine the critical value/s.
Compute the test statistic.
Decide the conclusion of the test.
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HYPOTHESIS TESTING
One Population
Two Populations
 One Mean
 Case 1
 Case 2
 One Proportion
 Difference of Two Means
 Case 1 to 3
 Paired Means
 Difference of Two
 One Variance
Proportions
 Difference of Two Variances
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Testing for One Mean
 Case 1: σ is known, or n ≥ 30
 Z-test will be used
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Testing for One Mean
 The test statistic is given by
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Example 1
 A manufacturer of sports equipment has developed a
new synthetic fishing line that he claims has a mean
breaking strength of 8 kilograms with a standard
deviation of 0.5 kilograms. Test the hypothesis that μ =
8 kg against the alternative that μ ≠ 8 kg if a random
sample of 50 lines is tested and found to have a mean
breaking strength of 7.8 kg. Use a 0.01 level of
significance.
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Identify the proper hypotheses
 The manager of the Danvers-Hilton Resort Hotel stated that the
mean guest bill for a weekend is $600 or less. A member of the
hotel’s accounting staff noticed that the total charges for guest
bills have been increasing in recent months. The accountant will
use a sample of weekend guest bills to test the manager’s claim.
 The manager of an automobile dealership is considering a new
bonus plan designed to increase sales volume. Currently, the
mean sales volume is 14 automobiles per month. The manager
wants to conduct a research study to see whether the new bonus
plan increases sales volume. To collect data on the plan, a sample
of sales personnel will be allowed to sell under the new bonus
plan for a one-month period.
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Identify the proper hypotheses
 A production line operation is designed to fill cartons with
laundry detergent to a mean weight of 32 ounces. A sample
of cartons is periodically selected and weighed to
determine whether underfilling or overfilling is occurring.
If the sample data lead to a conclusion of underfilling or
overfilling, the production line will be shut down and
adjusted to obtain proper filling.
 Because of high production-changeover time and costs, a
director manufacturing must convince management that a
proposed manufacturing method reduces costs before the
new method can be implemented. The current production
method operates with a mean cost of $220 per hour. A
research study will measure the cost of the new method
over a sample production
period.
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Case 2: σ unknown, AND n < 30
 t-test will be used
 The t-values have n – 1 degrees of freedom
 The test statistic is given by
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Example 2
 Test the hypothesis that the average content of
containers of a particular lubricant is 10 liters if the
contents of a random sample of 10 containers are 10.2,
9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9.8 liters. Use
a 0.01 level of significance and assume that the
distribution of contents is normal.
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Testing for One Proportion
 Z-test will be used
 The test statistic is given by
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Example 3
 A commonly prescribed drug on the market for
relieving nervous tension is believed to be only 60%
effective. Experimental results with a new drug
administered to a random sample of 100 adults who
were suffering from nervous tension showed that 70
received relief. Is this sufficient evidence to conclude
that the new drug is superior to the one commonly
prescribed? Use a 0.05 level of significance.
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Testing for One Variance
 A chi-square test will be used
 The test statistic is given by
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Example 4
 A manufacturer of car batteries claims that the life of
his batteries has a variance equal to 0.81 years. If a
random sample of 10 of these batteries have a variance
of 1.44 years, is there evidence that the variance
exceeds 0.81 a year? Use a 0.05 level of significance
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Testing the Difference of Two
Means
 Case 1:
 Z-test will be used
 The test statistic is given by:
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Example 5
 A manufacturer claims that the average tensile
strength of thread A exceeds the average tensile
strength of thread B by less than 12 kilograms. To test
this claim, 50 pieces of each type of thread are tested
under similar conditions. Type A thread had an
average tensile strength of 86.7 kilograms with a
standard deviation of 6.28 kilograms, while type B
thread had an average tensile strength of 77.8
kilograms with a standard deviation of 5.61 kilograms.
Test the manufacturer’s claim using a 0.05 level of
significance.
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Case 2: σ1=σ2 unknown, AND n1<30
and n2<30
 t-test will be used.
 The test statistic is given by
 The degrees of freedom to be used is
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Example 6
 A course in mathematics is taught to 12 students by the
conventional classroom procedure. A second group of
10 students was given the same course by means of
programmed materials. At the end of the semester the
same examination was given to each group. The 12
students meeting in the classroom made an average
grade of 85 with a standard deviation of 4, while the 10
students using programmed materials made an
average of 81 with a standard deviation of 5. Test the
hypothesis that the two methods of learning are equal
using a 0.10 level of significance. Assume the
population to be approximately normal with equal
variances. HT : PLBautista
Case 3: σ1≠σ2 unknown, AND n1<30
and n2<30
 t-test will be used
 The test statistic is given by
 The degrees of freedom is given by
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Example 7
 An improved manufacturing process is developed. The
quality-control tests show that the old process has an
average score of 12.8 with a standard deviation of 2.5
based on a sample of 8 observations, while the new
process shows an average score of 14.2 with a standard
deviation of 1.6 based on a sample of 10 observations.
Use a 0.05 level of significance to determine whether
there has been a significant increase in the average
scores of the new process, assuming unequal
variances.
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Paired Observations
 t-test will be used
 The test statistic is given by
 The degrees of freedom to be used is n – 1, where n is
the number of pairs
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Example 8
 To determine whether membership in a fraternity is
beneficial or detrimental to one’s grades, the
following grade-point averages were collected over a
period of 5 years:
 Assuming the populations to be normal, test at the
0.05 level of significance whether membership in a
fraternity is detrimental to one’s grades.
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Testing for Two Proportions
 Z-test will be used
 The test statistic is given by
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Example 9
 A vote is to be taken among the residents of a town
and the surrounding county to determine whether a
civic center will be constructed. To determine if there
is a significant difference in the proportion of town
voters and county voters favoring the proposal, a poll is
taken. If 120 of 200 town voters favor the proposal and
240 of 500 county residents favor it, would you agree
that the proportion of town voters favoring the
proposal is higher than the proportion of county
voters? Use a 0.025 level of significance.
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Testing for Two Variances
 F-test will be used
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Example 10
 Verify if the assumption of equal variance in Example
6 is valid by conducting a test of hypothesis. Use a 0.10
level of significance.
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p-value approach
 The p-value is the probability that we obtain the
sample data, assuming the null hypothesis is true.
 We reject Ho if the p-value is small.
 Usually, we use the level of significance as a comparison
value.
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Estimation and HT
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