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Confidence Intervals and Tests of Hypotheses for Means Study Guide (page 3)

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Updated on Oct 5, 2011

Step 4: Decide Whether or Not to Reject the Null Hypothesis

Before beginning the study, the significance level α of the test is set. If the p-value is less than the significance level, the null hypothesis is rejected; otherwise, the null is not rejected.

Step 5: State Conclusions in the Context of the Study

Statistical tests of hypotheses are conducted to determine whether or not sufficient evidence exists to reject the null hypothesis in favor of the alternative hypothesis.

Examples of Hypothesis Testing for a Mean

Below are two examples of Hypothesis Testing for a Mean using the five-steps previously discussed.

Example 1:

A machine is used to produce bolts that have a mean diameter of 1.50 cm. The diameters of the bolts are normally distributed. Sometimes, the machine produces bolts that differ from the desired 1.50 cm mean diameter. If the mean bolt diameter is significantly greater than 1.50 cm, the machine is to be reset. This is expensive because no bolts can be produced while it is being reset. A random sample of 15 bolts is selected from each day's production and tested, and their diameters are recorded. Today, the average diameter of the sampled bolts was 1.52 cm, and the sample standard deviation was 0.01 cm. Is there sufficient evidence to conclude that the machine needs to be reset?

Solution 1:

We follow the five steps of hypothesis testing.

Step 1: Specifying the Hypotheses

Here, we want strong evidence that the machine needs to be reset. This leads to the following set of hypotheses:

H0:μ = 1.50

Ha:μ > 1.50

Notice that here we have chosen μ = 1.50 for the null hypothesis because that is the mean diameter of bolts that the machine should be producing. We want to know if the mean diameter becomes significantly greater than 1.50, so μ > 1.50 becomes the alternative hypothesis.

Step 2: Verify Necessary Conditions for a Test and, if Satisfied, Construct the Test Statistic

The population distribution of bolt diameters is normal, and the bolts have been randomly selected from the day's production. Thus, the conditions for a test have been satisfied. The test statistic is:

= = 7.75

Step 3: Find the p-Value Associated with the Test Statistic

Assuming that the population distribution is normal and the standard deviation is known, the test statistic has a t-distribution with n – 1 = 14 degrees of freedom if the null hypothesis is true. We want to find the probability that a test statistic as extreme or more extreme would be observed if the null hypothesis is not true. Because we are interested only if the sample mean is too large (see the alternative hypothesis), we find

p = P(t > tT)

p = P(t > 7.75) = 1 – P(t ≤ 7.75) > 0.001

Step 4: Decide Whether or Not to Reject the Null Hypothesis

This p-value is so small that it will be less than any reasonable significance level. Therefore, we would reject the null hypothesis.

Step 5: State Conclusions in the Context of the Study

We reject the null hypothesis and conclude that the mean diameter of bolts produced on that day exceeds 1.50 cm. Therefore, the machine needs to be reset.

Example 2:

A researcher has read that U.S. children aged 10 to 17 years watch television an average of 3.6 hours per day. She wants to know if her region differs from this national average. She randomly selects 40 children aged 10 to 17 from the region. Because children do not watch exactly the same amount of television every day, she records the number of hours each sampled child watches television for a four-week period. By taking the average of the hours spent watching television over these 28 days, she has a good measure of the amount of time each child spends watching television daily. The sample mean number of hours these 40 children watched television daily is 2.8 hours, and the sample standard deviation is 2.2 hours. Is there sufficient evidence to conclude that the average number of hours children spend watching television in the researcher's region is different from that reported for the nation?

Solution 2:

Again, we follow the five steps of hypothesis testing.

Step 1: Specifying the Hypotheses

Here, we want to know whether the average for the region differs from that for the nation. The direction of that difference is not specified, so we have a two-sided alternative; that is, we want to test the following set of hypotheses:

H0:μ = 3.6

Ha:μ ≠ 3.6

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