Question

8. AM -vs- PM sections of Stats - Significance test (Raw Data, Software Required):
There are two sections of statistics, one in the afternoon (PM) with 30 students and one in the morning (AM) with 22 students. Each section takes the identical test. The PM section, on average, scored higher than the AM section. The scores from each section are given in the table below. 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) Use software to calculate the test statistic. Do not 'pool' the variance. This means you do not assume equal variances. Round your answer to 2 decimal places. t = (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.

PM Scores (x1) AM Scores (x2)    80 74 68 62 98 89 93 86 66 57 84 77 57 51 65 57 52 45 80 74 70 61 82 72 88 80 98 88 74 67 76 66 83 75 86 80 82 72 50 40 95 89 65 55 61 65 72 72 100 87 79 72

Answer 1

The analysis has been done with the help of R software.

(a). Test Statistic:

(b). P-value

(c). The conclusion regarding the null hypothesis is reject H0.

(d). 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

R code:

######## Two sample t-test #########
x1 = c(80, 68, 98, 93, 66, 84, 57, 65, 52, 80, 70, 82, 88, 98, 74, 76, 83, 86, 82, 50,
95, 65, 61, 65, 72, 72, 100, 87, 79, 72)
x2 = c(74, 62, 89, 86, 57, 77, 51, 57, 45, 74, 61, 72, 80, 88, 67, 66, 75, 80, 72, 40,
89, 55)

length(x1);length(x2)

t_test = t.test(x1,x2, alter = "greater", var.equal=TRUE)
t_test

######## The End ###################

Output:

> ######## Two sample t-test #########
> x1 = c(80, 68, 98, 93, 66, 84, 57, 65, 52, 80, 70, 82, 88, 98, 74, 76, 83, 86, 82, 50,
+ 95, 65, 61, 65, 72, 72, 100, 87, 79, 72)
> x2 = c(74, 62, 89, 86, 57, 77, 51, 57, 45, 74, 61, 72, 80, 88, 67, 66, 75, 80, 72, 40,
+ 89, 55)
>
> length(x1);length(x2)
[1] 30
[1] 22
>
> t_test = t.test(x1,x2, alter = "greater", var.equal=TRUE)
> t_test

Two Sample t-test

data: x1 and x2
t = 2.0001, df = 50, p-value = 0.02547
alternative hypothesis: true difference in means is greater than 0
95 percent confidence interval:
1.249937 Inf
sample estimates:
mean of x mean of y
76.66667 68.95455

>
> ######## The End ###################
>
>