Question

1.) Ashley, of Bhak Dat Stat Up! (a Tsygylyk subsidiary), wanted to test the claim that the standard deviation for the number of months in a relationship, before a couple had their first argument, would be at least 5 months, She randomly surveyed forty couples and found their mean to be 12.9 months with a standard deviation of 3.1 months. At a 5% significance level, can he conclude that the standard deviation for all such couples is at least 5 months?

2.) D-Bar! Magazine wanted to see if a statistics tutorial class had any effect on college quiz scores; total possible points were 25. They sent , of “Rasta Mon!” fame, to survey a sample of six statistics students that took this tutorial class. The table below gives their scores before and after the completion of the “stats” tutorial class. (Assume an approx. normal distribution.)

Before	12 18 25 9 14 16
After	18 24 24 14 19 20

Construct a 95% confidence interval for the mean differences between the scores before and after the completion of the “stats” tutorial class.

Answer 1

1.

We have to perform one sample Chi-square test for population variance.

Here population mean is unknown. So we have to use sample mean and this decreases degrees of freedom by one from sample size.

We have to test for null hypothesis
against the alternative hypothesis

Our test statistics is given by

Here,

Sample size

Sample variance

Degrees of freedom

[Using R-code '1-pchisq(14.9916,39)']

Level of significance

We reject our null hypothesis if

Here, we observe that

So, we cannot reject our null hypothesis.

Hence, based on the given data we can conclude that there is not significant evidence that population standard deviation is at least 5 months.

2.

Here scores of same statistics students in two different instances were observed. So, we have to use paired t-test statistic.

Suppose, random variables X and Y denote scores before and after the tutorial class respectively. Also, random variable D(=Y-X) denotes difference in scores.

Before class (X)	12	18	25	9	14	16
After class (Y)	18	24	24	14	19	20
D=Y-X	6	6	-1	5	5	4

Corresponding statistic is given by

Here,

Number of pair of samples

Degrees of freedom

We know,

[Using R-code 'qt(1-(1-0.95)/2,5)']

Hence, 95% confidence interval for the mean differences between the scores before and after the completion of the “stats” tutorial class is given by (1.396739, 6.936595).