Question

Carry probabilities to at least four decimal places for intermediate steps. For extremely small probabilities, it is important to have one or two significant non-zero digits, for example, 0.000001 or 0.000034. Round off your final answer to two decimal places.

Benign prostatic hyperplasia is a noncancerous enlargement of the prostate gland that adversely affects the quality of life of millions of men. A study of a minimally invasive procedure for the treatment for this condition looked at pretreatment quality of life (QOL_BASE) and quality of life after three months on treatment (QOL_3MO). The data is shown on the following table:

Participant	QOL_BASE	QOL__3MO
1	2	1
2	4	1
3	3	1
4	4	3
5	5	2
6	6	2
7	4	2
8	4	5
9	3	3
10	3	1

a)Calculate differences in quality of life scores (delta) for each participant. (1 point)

b)Calculate the means for QOL_BASE, QOL_3MO, and the delta (difference in QOL_BASE and QOL_3MO). (3 points)

c)Calculate the standard deviation of the delta (difference in QOL_BASE and QOL_3MO. (1 point)

d)Test the mean difference for statistical significance. Use a two-sided alternative test and ? = 0.05. Include all hypotheses testing steps for full points. (Hint: Paired sample) (10 points)

e)Calculate a 95% confidence interval, and provide an interpretation. Does your interval further support your decision in the hypothesis test above? (5 points)

Answer 1

a) Calculate differences in quality of life scores (delta) for each participant. The following table shows the difference

Participant	QOL_BASE	QOL__3MO	Delta
1	2	1	1
2	4	1	3
3	3	1	2
4	4	3	1
5	5	2	3
6	6	2	4
7	4	2	2
8	4	5	-1
9	3	3	0
10	3	1	2

b) Let the random variables be QOL_BASE, be QOL_3MO and be the difference in  QOL_BASE and QOL_3MO

The sample ,means are

c) The standard deviation of the delya is

d) Let be the true mean difference in QOL_BASE and QOL_3MO. We want to test if there is a difference in mean quality of life score, or we want to test if

The following are the hypotheses

Since the same set of patients are tracked for before and after scores, these samples are dependent or paired samples.

Since the sample size n=10 is less than 30 and we do not know the population standard deviation, we will use t statistics.

We estimate the population standard deviation of the difference using sample standard deviation

the standard error of mean difference is

The hypothesized value of the mean difference is zero, or

the t staatistics is

The degrees of freedom is 10-1 = 9

This is a 2 tailed test. The critical value of T for alpha = 0.05 is given by

The critical value of t is +-2.262

Since the sample t statistics of 3.60 is outside the region of acceptance (-2.262,+2.262), we reject the null hypothesis.

We conclude that there is sufficient evidence to the claim that the mean difference  in quality of life scores is not equal to zero.

or we conclude that here is sufficient evidence to the claim that the mean quality of life scores before and after treatment are different.

e) 95% confidence interval indicates that the total area under the 2 tails is 0.05. Or the area under each tail is 0.025. The area under the right tail is

. For degrees of freedom 9 we get

The 95% confidence interval is

The hypothesized value of the mean difference is zero. This hypothesized value does not fall within the confidence interval. That means we can be 95% certain that the true mean difference is not zero.

The conclusion is the same as what we arrived at using hypothesis testing. We conclude that there is sufficient evidence to the claim that the mean difference is not equal to zero.