Confidence Intervals and Tests of Hypotheses for Means Study Guide (page 4)

Updated on Oct 5, 2011

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

A random selection of children was taken from all children in the region (the population of interest). The population distribution of the number of hours children spend watching television daily is very unlikely to be normal. However, because the sample size is 40 (> 30), we know that has at least an approximate normal distribution by the Central Limit Theorem. Therefore, the two conditions for inference are satisfied. The test statistic is

= = 1.15.

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

The test statistic has an approximate t-distribution with n – 1 = 39 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 if the sample mean gets "too far" from the hypothesized population mean, we f i n d p =P(|t| > |tT|) = p =P(|tT| > 1.15) > 2 × 0.1 = 0.2 using the table in Lesson 12. Notice that the smallest t-value in the line for 39 degrees of freedom in the t-table is 1.304, corresponding to a right-tail probability of 0.01. Because 1.15 is smaller than 1.304, more area is to the right of 1.15 than to the right of 1.304. Thus, P(t > 1.15) > 0.1.Because the alternative hypothesis is two sided, this right-tail probability must be doubled to account for the values in the left-tail that would have also been at least as extreme as what we observed if the null hypothesis is true.

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

The p-value being greater than 0.2 indicates that what we observed would not be at all unusual if the null hypothesis is true (p > α).Therefore, we would not reject the null hypothesis.

Step 5: State Conclusions in the Context of the Study

We fail to reject the null hypothesis that the mean number of hours children in the researcher's region watch television is the same as that for children in the nation.

Confidence Intervals and Tests of Hypotheses for Means In Short

The sample mean is the best estimate of the population mean, but the two are rarely equal. Thus, we use confidence intervals to establish an interval of values that will capture the true population mean with a specified level of confidence. The form of these intervals continues to be estimate ± multiplier× standard error as it was for proportions. We may also want to test hypotheses concerning values of the population mean. The test statistic for means is , where μ0 is the hypothesized value of the population mean.

Find practice problems and solutions for these concepts at Confidence Intervals and Tests of Hypotheses for Means Practice Questions.

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