Question

In general, high school and college students are the most pathologically sleep-deprived segment of the population. Their alertness during the day is on par with that of untreated narcoleptics and those with untreated sleep apnea. Not surprisingly, teens are also 71 percent more likely to drive drowsy and/or fall asleep at the wheel compared to other age groups. (Males under the age of twenty-six are particularly at risk.)

The accompanying data set represents the number of hours 25 college students at a small college in the northeastern United States slept and is from a random sample. Enter this data into C1 of Minitab Express.

6 9 7 7 6 7 7 5 8 6 6 6 8 8 8 5 4 6 7 8 5 8 7 6 7

For the analyses that follow, we shall use

·         90%, 95%, and 99% as the confidence levels for the confidence interval.

·      5% as the level of significance ( ) for the hypothesis test.

·         7 hours sleep as the null hypothesis (according to The Sleep Foundation).

i.       Write the null ( ) and alternative ( ) hypotheses to test if the sample data provides sufficient evidence to support the claim that the mean number of hours slept is less than 7 hours.

a.    Manually calculate the observed value of the test statistic, . First, write the symbolic equation, then replace the inputs with their respective values, and finally calculate the numeric value.

a.    Manually calculate the observed value of the test statistic, . First, write the symbolic equation, then replace the inputs with their respective values, and finally calculate the numeric value.

Answer 1

From the minitab the output for Confidence intervals for mean sleep hours are as follows

Descriptive Statistics

N	Mean	StDev	SE Mean	90% CI for μ	95% CI for μ 99% CI for μ
25	6.680	1.215	0.243	(6.264, 7.096)	(6.178, 7.182)(6.000, 7.360)

μ: mean of Sleep Hours

Hypothesis:

H₀ : = 7 ( The sleep hours of students is 7 hours)

H₁ : < 7 (The mean sleep hours of students is less than 7 hours)

level of significance = 5%

test statistic t =

where = mean of the sample, s= standard deviation of the sample, n = sample size and = population mean.

t = = -1.32

critical value for 0.05 level of significance and n-1 degrees of freedom

t_0.05,24 = 1.711

since t is not less than - t_0.05,24 we faill to reject null hypothesis. We conclude that statistically there is no evidence to claim that sleep hours is less than 7hours.