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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
OBJECTIVES
1. Test a hypothesis about a mean using the P-value method
2. Test a hypothesis about a mean using the critical value method
OBJECTIVE 1
TEST A HYPOTHESIS ABOUT A MEAN USING THE P-VALUE METHOD
We begin this section with an example. Suppose that in a recent medical study, 76 subjects
were placed on a low-fat diet. After 12 months, their sample mean weight loss was 𝑥̅ = 2.2
kilograms, with a sample standard deviation of 𝑠 = 6.1 kilograms. Can we conclude that the
mean weight loss is greater than 0?
If we knew the population standard deviation 𝜎, we would be able to compute the 𝑧-score of
𝑥̅ −𝜇
the sample mean to be 𝑧 = 𝜎/ 𝑛, and use this test statistic to perform a hypothesis test. In this
√
example, as is usually the case, we do not know the population standard deviation. To proceed,
𝑥̅ −𝜇
we replace 𝜎 with the sample standard deviation 𝑠, and use the 𝑡 test statistic instead: 𝑡 = 𝑠/ 𝑛.
√
When the null hypothesis is true, the 𝑡 statistic has a Student’s 𝑡 distribution with 𝑛 − 1
degrees of freedom.
The assumptions for performing a hypothesis test for 𝜇 when the population standard deviation
𝜎 is unknown are as follows:
1. We have a simple random sample.
2. The sample size is large (𝑛 > 30), or the population is approximately normal.
Since we have a simple random sample and the sample size is large, we may proceed with the
test. The issue is whether the mean weight loss 𝜇 is greater than 0. So the null and alternate
hypotheses are 𝐻0 : 𝜇 = 0 versus 𝐻1 : 𝜇 > 0.
𝑥̅ −𝜇
The test statistic is 𝑡 = 𝑠/
√𝑛
2.2−0
= 6.1/√76 = 3.144. When 𝐻0
is true, the test statistic 𝑡 has the Student’s 𝑡 distribution
with 𝑛 − 1 = 76 − 1 = 75 degrees of freedom. This is a
right tail test, so the P-value is the area under the Student’s
𝑡 curve to the right of 𝑡 = 3.144. Using technology, we find
the exact P-value to be P = 0.0012.
Since P < 0.05, we reject 𝐻0 at the 𝛼 = 0.05 level. We conclude that the mean weight loss of
people who adhered to this diet for 12 months is greater than 0.
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
COMPUTING P-VALUES
The P-value of the test statistic 𝑡 is the probability, assuming 𝐻0 is true, of observing a value for
the test statistic that disagrees as strongly as or more strongly with 𝐻0 than the value actually
observed. The P-value is an area under the Student’s 𝑡 curve with 𝑛 − 1 degrees of freedom.
The area is in the left tail, the right tail, or in both tails, depending on the type of alternate
hypothesis.
ESTIMATING THE P-VALUE F ROM A TABLE
When using a 𝑡 table, we cannot find the P-value exactly. Instead, we can only specify that P is
between two values. In the last example, there are 75 degrees of freedom. We consult Table
A.3 and find that the number 75 does not appear in the degrees of freedom column. We
therefore use the next smallest number, which is 60. Now look across the row for two numbers
that bracket the observed value 3.144. These are 2.915 and 3.232. The upper-tail probabilities
are 0.0025 for 2.915 and 0.001 for 3.232. The P-value must therefore be between 0.001 and
0.0025. We can conclude that the P-value is small enough to reject 𝐻0 at the 𝛼 = 0.05 level of
significance.
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
P-VALUE FROM A TABLE FOR A TWO-TAILED TEST
If the alternate hypothesis were 𝐻1 : 𝜇 ≠ 0, the P-value would be the sum of the areas in two
tails. If using Table A.3, we can only specify that P is between two values. We know, from
Table A.3, that the area in one tail is between 0.001 and 0.0025. Therefore, the area in both
tails is between 2(0.001) = 0.002 and 2(0.0025) = 0.005.
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
E XAMPLE :
Generic drugs are lower-cost substitutes for brand-name drugs. Before a generic
drug can be sold in the United States, it must be tested and found to perform
equivalently to the brand name product. The U.S. Food and Drug Administration
is now supervising the testing of a new generic antifungal ointment. The brandname ointment is known to deliver a mean of 3.5 micrograms of active
ingredient to each square centimeter of skin. As part of the testing, seven
subjects apply the ointment. Six hours later, the amount of drug that has been
absorbed into the skin is measured. The amounts, in micrograms, are
2.6
3.2
2.1
3.0
3.1
2.9
3.7
How strong is the evidence that the mean amount absorbed differs from 3.5
micrograms? Use the 𝛼 = 0.01 level of significance.
S OLUTION :
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
HYPOTHESIS TESTING ON THE TI-84 PLUS
The TTest command will perform a hypothesis test when
the population standard deviation 𝜎 is not known. This
command is accessed by pressing STAT and highlighting the
TESTS menu.
If the summary statistics are given the Stats option should
be selected for the input option.
If the raw sample data are given, the Data option should be
selected.
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
OBJECTIVE 2
TEST A HYPOTHESIS ABOUT A MEAN USING THE CRITICAL VALUE METHOD
CRITICAL VALUES FOR THE 𝒕-STATISTIC
The critical value method for a hypothesis test of a population mean when 𝜎 is unknown is the
same as that when 𝜎 is known. The only exception is that we use the Student’s 𝑡 distribution
rather than a normal distribution. The critical values for the Student’s 𝑡 distribution can be
found in Table A.3 or with technology.
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
E XAMPLE :
A computer software vendor claims that a new version of their operating system
will crash less than six times per year on average. A system administrator installs
the operating system on a random sample of 41 computers. At the end of a
year, the sample mean number of crashes is 7.1, with a standard deviation of
3.6. Can you conclude that the vendor’s claim is false? Use the 𝛼 = 0.05
significance level.
S OLUTION :
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SECTION 9.3: HYPOTHESIS TESTS FOR A
POPULATION MEAN, 𝜎 UNKNOWN
YOU SHOULD KNOW …

The assumptions for hypothesis tests for 𝜇 when 𝜎 is unknown

How to perform hypothesis tests for 𝜇 when 𝜎 is unknown using the P-value method

How to estimate a P-value from a table for one-tailed and two-tailed tests

How to perform hypothesis tests for 𝜇 when 𝜎 is unknown using the critical value
method
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