406 CHAPTER 8 Hypothesis Testing Important Properties of the Student t Distribution Here is a brief review of important properties of the Student t distribution first presented in Section 7-2: 1. The Student t distribution is different for different sample sizes (see Figure 7-4 in Section 7-2). 2. The Student t distribution has the same general bell shape as the standard normal distribution; its wider shape reflects the greater variability that is expected when s is used to estimate s. 3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0). 4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has s = 1). 5. As the sample size n gets larger, the Student t distribution gets closer to the standard normal distribution. P-Value Method with Technology If suitable technology is available, the P-value method of testing hypotheses is the way to go. Adult Sleep: P-Value Method with Technology EXAMPLE 1 The author obtained times of sleep for randomly selected adult subjects included in the National Health and Nutrition Examination Study, and those times (hours) are listed below. Here are the unrounded statistics for this sample: n = 12, x = 6.83333333 hours, s = 1.99240984 hours. A common recommendation is that adults should sleep between 7 hours and 9 hours each night. Use the P-value method with a 0.05 significance level to test the claim that the mean amount of sleep for adults is less than 7 hours. 4 8 4 4 8 6 9 7 7 10 7 8 SOLUTION REQUIREMENT CHECK (1) The sample is a simple random sample. (2) The second requirement is that “the population is normally distributed or n 7 30.” The sample size is n = 12, which does not exceed 30, so we must determine whether the sample data appear to be from a normally distributed population. The accompanying histogram and normal quantile plot, along with the apparent absence of outliers, indicate that the sample appears to be from a population with a distribution that is approximately normal. Both requirements are satisfied. Statdisk
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