Question

9.

There are two sections of statistics, one in the morning (AM) with 20 students and one in the afternoon (PM) with 31students. Each section takes the identical test. The PM section, on average, scored higher than the AM section. The results are summarized in the table below.

Necessary information:

	n	x	s2	s
PM (x1)	31	81.4	277.5	16.66
AM (x2)	20	71.5	250.3	15.82

The Test: Test the claim that the PM section did significantly better than the AM section, i.e., is the difference in mean scores large enough to believe that something more than random variation produced this difference. Use a 0.05 significance level.

(a) Calculate the test statistic using software or the formula below
t =

(x₁ − x₂) − δ

s12 n1 + s22 n2

where δ is the hypothesized difference in means from the null hypothesis. Round your answer to 2 decimal places.

t =

  
To account for hand calculations -vs- software, your answer must be within 0.01 of the true answer.  

(b) Use software to get the P-value of the test statistic. Round to 4 decimal places.
P-value =  

(c) What is the conclusion regarding the null hypothesis?

reject H₀fail to reject H₀     

(d) Choose the appropriate concluding statement.

The difference in mean scores is large enough to suggest this difference is due to something more than random variation. There is not a big enough difference in mean scores to suggest that this difference is anything more than a result of random variation.      We have proven that students in PM sections of statistics do better, on average, than students taking AM sections.We have proven that there is no difference between AM and PM sections of statistics.

Answer 1

For PM : x̅1 = 81.4, s1 = 16.66, n1 = 31

For AM : x̅2 = 71.5, s2 = 15.82, n2 = 20

Null and Alternative hypothesis:

Ho : µ1 = µ2

H1 : µ1 > µ2

a)

Test statistic:

t = (x̅1 - x̅2)/√(s1²/n1 + s2²/n2) = (81.4 - 71.5)/√(16.66²/31 + 15.82²/20) = 2.14

b)

df = ((s1²/n1 + s2²/n2)²)/[(s1²/n1)²/(n1-1) + (s2²/n2)²/(n2-1) ] = 42.2251 = 42

p-value = T.DIST.RT(2.1367, 42) = 0.0192

c)

p-value < α, Reject the null hypothesis

d)

Conclusion:

We have proven that students in PM sections of statistics do better, on average, than students taking AM sections.