Question

Is the average amount of time spent sleeping each day different between male and female students?

1. Which hypothesis test is the most appropriate to use in this problem? Why? (Note: if you are doing a 2-sample t-test, make sure you state which one you are doing and why.) If the test you chose has a t-statistic, report it here with degrees of freedom. If it does not, state that the test you chose does not have a test-statistic. Give and interpret an appropriate 95% confidence interval for the difference in population mean time spent sleeping each day between male and female students.

sex	sleep
Female	540
Male	420
Female	450
Female	420
Female	480
Female	420
Male	400
Male	420
Female	480
Female	480
Male	360
Female	500
Female	270
Female	480
Male	540
Female	420
Male	360
Male	420
Male	480
Male	480
Male	420
Male	560
Male	280
Male	570
Male	480
Male	480
Male	420
Male	420
Male	480
Male	360
Male	480
Male	480
Female	360
Male	480
Male	480
Female	480
Male
Female	480
Male	465
Male	500
Male	420
Male	420
Female	600
Female	480
Female	420
Female	480
Female	480
Male	480
Female	480
Female	480
Female	300
Female	540
Female	420
Male	480
Female	480
Female	360
Female	480
Male	480
Male	450
Male	420
Male	480
Male
Female	540
Female	420
Male	450
Male	360
Female	480
Male	360
Male	540
Female	480
Male	420
Male	480
Male	240
Female	480
Male	420
Female	420
Female	450
Female	380
Male	540
Female	480
Female
Male
Male	480
Male	420
Female	420
Male	360
Female	540
Male	420
Male	480
Male	420
Female	450
Male	480
Female
Female	480
Male	560
Male	480
Male	540
Male	420
Male	480
Male	480
Female	450
Female	480
Female	450
Female	420
Female	480
Male	420
Female	540
Female	480
Male
Male	360
Female	720
Female	480
Male	300
Male	360
Male	420
Female	420
Male	540
Male	480
Female	480
Male	420
Male	420
Male	480
Female	360
Female	460
Female	480
Male	420
Female	420
Male	480
Male	360
Male	480
Female	420
Female	420
Female	490
Female	450
Male	460
Male	540
Male	450
Male	400
Female	360
Male	420
Male	500
Female	420
Male	390
Male	450
Male	500
Female	480
Male	540
Male	480
Male	540
Female	480
Male	480
Male
Female	500
Female	600
Male	540
Male	480
Female	600
Female	420
Female	480
Male	420
Male	480
Female	300
Male	420
Male	400
Female	330
Male	390
Male
Male	480
Female	480
Male	450
Male	420
Male	420
Male	480
Female	480
Female	450
Female	480
Female	420
Male	420
Male	480
Female	390
Female	360
Male	450
Male	480
Female	480
Female	480
Male	380
Male	360
Female	480
Male	420
Female	420
Male	360
Male	440
Male
Male	390
Female	420
Female	720
Female	480
Female	480
Female	480
Female	420
Female	360
Male	480
Male	420
Male	400
Male	480
Female	420
Male	480
Female	420
Female	480
Male	300
Male	480
Female	300
Female	450
Female	480
Male	450

Answer 1

(a) Hypothesis test:

Since the population standard deviation is not known, we conduct t- test for independent samples

Female

n = 94

x-bar = 457.23

s = 71.85

Male

n = 112

x-bar = 444.96

s = 60.55

Data:

n1 = 94

n2 = 112

x1-bar = 457.23

x2-bar = 444.96

s1 = 71.85

s2 = 60.55

Hypotheses:

Ho: μ1 = μ2

Ha: μ1 ≠ μ2

Decision Rule:

α = 0.05

Degrees of freedom = 94 + 112 - 2 = 204

Lower Critical t- score = -1.971660843

Upper Critical t- score = 1.971660843

Reject Ho if |t| > 1.971660843

Test Statistic:

Pooled SD, s = √[{(n1 - 1) s1^2 + (n2 - 1) s2^2} / (n1 + n2 - 2)] =   √(((94 - 1) * 71.85^2 + (112 - 1) * 60.55^2)/(94 + 112 -2)) =    65.94

SE = s * √{(1 /n1) + (1 /n2)} = 65.9420746252524 * √((1/94) + (1/112)) = 9.224084622

t = (x1-bar -x2-bar)/SE = 1.330213295

p- value = 0.184933196

Decision (in terms of the hypotheses):

Since 1.330213295 < 1.971660843 we fail to reject Ho

Conclusion (in terms of the problem):

There is no sufficient evidence that the mean sleeping times are different

(b) confidence interval:

n1 = 94

n2 = 112

x1-bar = 457.23

x2-bar = 444.96

s1 = 71.85

s2 = 60.55

% = 95

Degrees of freedom = n1 + n2 - 2 = 94 + 112 -2 = 204

Pooled s = √(((n1 - 1) * s1^2 + (n2 - 1) * s2^2)/DOF) = √(((94 - 1) * 71.85^2 + ( 112 - 1) * 60.55^2)/(94 + 112 -2)) = 65.94207463

SE = Pooled s * √((1/n1) + (1/n2)) = 65.9420746252524 * √((1/94) + (1/112)) = 9.224084622

t- score = 1.971660843

Width of the confidence interval = t * SE = 1.97166084255928 * 9.22408462239903 = 18.18676646

Lower Limit of the confidence interval = (x1-bar - x2-bar) - width = 12.27 - 18.1867664584374 = -5.916766458

Upper Limit of the confidence interval = (x1-bar - x2-bar) + width = 12.27 + 18.1867664584374 = 30.45676646

The 95% confidence interval is [-5.9168, 30.4568]

Interpretation:

If samples of size 94 and 112 are repeatedly drawn from the population and the confidence intervals for the difference between their means constructed such intervals will contain the true mean difference 95% of the time.