Hypothesis Testing. In Exercises 17–19, apply the central limit theorem to test the given claim. (Hint: See Example 3.)


Adult Sleep Times (hours) of sleep for randomly selected adult subjects included in the National Health and Nutrition Examination Study are listed below. Here are the statistics for this sample: n = 12, x_bar = 6.8 hours, s = 20 hours. The times appear to be from a normally distributed population. A common recommendation is that adults should sleep between 7 hours and 9 hours each night. Assuming that the mean sleep time is 7 hours, find the probability of getting a sample of 12 adults with a mean of 6.8 hours or less. What does the result suggest about a claim that “the mean sleep time is less than 7 hours”?


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Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μ = 7 hours (the mean sleep time is 7 hours). The alternative hypothesis is H₁: μ < 7 hours (the mean sleep time is less than 7 hours).
2. Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is SE = s / √n, where s is the sample standard deviation and n is the sample size. Substitute the given values: s = 20 and n = 12.
3. Step 3: Compute the test statistic (z-score). The formula for the z-score is z = (x̄ - μ) / SE, where x̄ is the sample mean, μ is the population mean, and SE is the standard error. Substitute the given values: x̄ = 6.8, μ = 7, and the SE calculated in Step 2.
4. Step 4: Find the p-value corresponding to the calculated z-score. Use a standard normal distribution table or statistical software to determine the probability of observing a z-score less than the calculated value.
5. Step 5: Compare the p-value to the significance level (commonly α = 0.05). If the p-value is less than α, reject the null hypothesis. Interpret the result in the context of the problem: Does the evidence suggest that the mean sleep time is less than 7 hours?

Key Concepts

Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the samples are independent and identically distributed. This theorem is crucial for hypothesis testing, as it allows us to make inferences about population parameters based on sample statistics, especially when dealing with small sample sizes.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0 in favor of H1. In this context, the null hypothesis would state that the mean sleep time is 7 hours, while the alternative would suggest it is less than 7 hours.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis in hypothesis testing. It represents the probability of obtaining a sample mean as extreme as, or more extreme than, the observed sample mean, assuming the null hypothesis is true. A low P-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that the mean sleep time may indeed be less than 7 hours.