Hypothesis Testing. In Exercises 17–19, apply the central limi...

Question

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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Accepted Answer

All right, hello, everyone. So this question says, a study suggests that the average commute time for employees in the city is 35 minutes. A random sample of 10 employees shows a mean commute time of 33 minutes with a sample standard deviation of 5 minutes. Assume the commute times are normally distributed. What is the probability of getting a sample mean of 33 minutes or less if the true mean is 35 minutes? What does this result suggest about the claim that the mean commute time is less than 35 minutes? So first, we want to state the hypotheses that are relevant to this question. In this case, we can have a null hypothesis or H0. And an alternative hypothesis or HA. So the null hypothesis would state that mu is equal to 35 minutes. Which means that the true mean is 35 minutes. The alternative hypothesis, on the other hand, would say that mew is less than 35 minutes. Meaning that the true mean is less than 35 minutes. So, because the population standard deviation is not known at this time, and the sample size is relatively small, we can use the T test. Where tea Oops, T is equal to X bar, that's the sample mean, subtracted by mu of the null hypothesis, divided by. The standard deviation divided by the square root of the sample size. So plugging in the information we talked about earlier. T is equal to 33. Subtracted by 35. And then divided by 5 divided by the square root of 10. Evaluating this expression gives you a T value. Equal to approximately 1.265. So now with T equal to -1.265, let's find the degrees of freedom. Which is equal to and subtracted by 1. So 10 subtracted by 1 equals 9 degrees of freedom. So now with this information in mind, we can now find the probability or the p-value. In this case, T is equal to -1.265, but recalled the T distribution is symmetrical. So we can treat this as the absolute value of tea. Being equal to 1.265 when using the tea table. So here, 4 degrees of freedom equal to 9, the critical values in the tea table. R T equals 1.383. 4 P equals negative. Excuse me, 0.10. And T equals 1.833. For P equals 0.05. So here, we're going to have to extrapolate the P value for T equals 1.265, since it's between these two from the T table. So using linear extrapolation. Let me actually scroll down to open up some more space here. As mentioned before, using linear extrapolation with the given data. We can say that P subtracted by 0.05. Divided by 0.10, subtracted by 0.05, equals 1.265. Subtracted by 1.833. Divided by 1.383. Subtracted by 1.833. This means that P is equal to. 1.265, subtracted by 1.833. Divided by 1.383, subtracted by 1.833. Multiplied by 0.10, subtracted by 0.05. And then added to. 0.05. And so after evaluating this expression, P is equal to approximately 0.113. So now, how can we interpret the result that we just obtained? A P value equal to 0.112, or approximately 0.113, means that the probability of getting a sample mean of 33 minutes. Or less Is 0.113. Now, our P value is actually greater than 0.05, which means that it is not small enough to be able to reject the null hypothesis, meaning there is not enough evidence to support the claim that the mean commute time is truly less than 35 minutes. Which leads us now to our final answer that I've pre-written here on the screen. Here, P is equal to 0.112. This suggests there is not enough evidence to claim the mean commute time is less than 35 minutes. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Central Limit Theorem

Hypothesis Testing

P-value

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