The National Sleep Foundation surveyed representative samples of adults in six different countries to ask
questions about sleeping habits. Each person in a representative sample of 250 adults in each of these
countries was asked how much sleep they get on a typical work night. For the United States, the sample
mean was 391 minutes, and for Mexico the sample mean was 426 minutes. Suppose that the sample
standard deviations were 23 minutes for the U.S. sample and 48 minutes for the Mexico sample. The report
concludes that on average, adults in the United States get less sleep on work nights than adults in Mexico.
Is this a reasonable conclusion? Support your answer with an appropriate hypothesis test. (Use alpha = 0.05
Use 𝜇1 for Mexico and 𝜇2 for the United States.)
a) State the appropriate null and alternative hypotheses
b) Find the test statistic. (Round your answer to two decimal places.)
c) Use technology to find the P-value.
Guide On Rating System
Vote
a) The appropriate null and alternative hypotheses are:
Null Hypothesis (H0): The average amount of sleep on work nights is the same for adults in the United States and adults in Mexico.
Alternative Hypothesis (H1): The average amount of sleep on work nights is different for adults in the United States and adults in Mexico.
b) The test statistic can be found using the following formula:
\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]
where:
- \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means for the United States and Mexico respectively
- \(s_1\) and \(s_2\) are the sample standard deviations for the United States and Mexico respectively
- \(n_1\) and \(n_2\) are the sample sizes for the United States and Mexico respectively
Given:
- \(\bar{x}_1 = 391\)
- \(\bar{x}_2 = 426\)
- \(s_1 = 23\)
- \(s_2 = 48\)
- \(n_1 = n_2 = 250\)
Now, let's calculate the test statistic:
\[ t = \frac{391 - 426}{\sqrt{\frac{23^2}{250} + \frac{48^2}{250}}} \]
c) To find the P-value using technology, we need to find the degrees of freedom and then use the t-distribution.
The degrees of freedom can be calculated using the following formula:
\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}} \]
Plug in the given values:
\[ df = \frac{\left(\frac{23^2}{250} + \frac{48^2}{250}\right)^2}{\frac{\left(\frac{23^2}{250}\right)^2}{250-1} + \frac{\left(\frac{48^2}{250}\right)^2}{250-1}} \]
Once we have the degrees of freedom, we can use technology (such as online t-distribution calculators or software) to find the P-value.
Got a question with my answer?
Cheapmath.com is completely free!
Asking question is free.
Getting answers is free.
+99