Carefully read each question. Clearly label all answers listing formulas as appr
ID: 3227902 • Letter: C
Question
Carefully read each question. Clearly label all answers listing formulas as appropriate. All work must be shown for partial credit. Please round your calculations to four decimal places. When completing a test of significance use the four-step process.
In a Sleep in America survey of American teens conducted by the National Sleep Foundation, teens were found to sleep an average of 6.5 hours per school night (according to research, they actually need more than 9 hours). In a recent study on 40 teens the average hours of sleep per night was found to be 6. Assume that the population is normally distributed with a standard deviation of one hour.
a. Is there statistically significant evidence at the 1% level of significance, that the average number of hours of sleep per school night is not 6.5?
b. Calculate a 99% confidence interval for . Does this interval support your results of part a? Why or why not?
c. If the research claimed that the number of hours of sleep for teens was less than 6.5 hours, then how does the p-value change? Is this stronger or weaker evidence than that of part a? Explain your answer.
d. Would the evidence in part a be stronger or weaker if the sample size were increased, assuming everything else remained the same? Explain your answer.
Explanation / Answer
a. Test Statistic : Mean of the sample xbar = 6 Hr and standard deviation s =1 Hr and H = 6.5 hr
t = ( - Xbar)/ (s/n) = (6.5 - 6)/ (1/40) = 0.5/ 0.15811 = 3.1623
so for 99% confidence interval and dF = 39
tcritical = 2.708
The P-Value is .003027.The result is significant at p < .01.
so t > tcritical so we can reject the null hypothesis and can say that the average number of hours of sleep per school night is not 6.
(b) 99% confidence interval for population mean .
99% confidence interval = xbar +- tcritical (s/n)
= 6 +- 2.708 * (1/40) = (5.57, 6.43)
so we support the results of part a.
(c) If the research claimed that the number of hours of sleep for teens was less than 6.5 hours.
So NUll HYpothesis is that sleep for theen was less than 6.5 Hours then it will be a one tailed test.
Where tcritical = 2.425 (H <= 6.5)
The P-Value is .001514. The result is significant at p < .01.
Here P - value is less than the case in part(a). It is stronger evidence than that of part a.
(d) If sample size were increased, evidence in part a will be stronger because it is inversely proportion to sample size.