Question

A state legislator wants to determine whether his voters' performance rating (0 - 100) has changed from last year to this year. The following table shows the legislator's performance from the same ten randomly selected voters for last year and this year. Use this data to find the 80% confidence interval for the true difference between the population means.

Let d=(this year's rating)−(last year's rating). Assume that the populations of voters' performance ratings are normally distributed for both this year and last year.

Rating (last year)	Rating (this year)
86	92
84	77
47	69
44	51
84	64
88	72
57	87
44	56
51	43
67	69

step 1:

Find the mean of the paired differences, d‾. Round your answer to one decimal place.

Step 2:

Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Step 3:

Find the standard deviation of the paired differences to be used in constructing the confidence interval. Round your answer to one decimal place.

Step 4:

Construct the 95% confidence interval. Round your answers to one decimal place.

Answer 1

Step 1

Rating (last year)	Rating (this year)	Paired difference (d)
86	92	6
84	77	-7
47	69	22
44	51	7
84	64	-20
88	72	-16
57	87	30
44	56	12
51	43	-8
67	69	2
	Total difference	28
	mean difference	2.8

Mean of the paired differences = 2.8

Step 2

The t-critical value is 1.383 for 80% Confidence interval

The t-critical value is  1.833 for 95% Confidence interval

Step 3

Standard deviation =16.0

Step 4:

The confidence interval is given by :

80% CI is

= (-4.2, 9.8)

95% CI is

= (-6.5, 12.1)

Thus the 80% confidence interval is (-4.2, 9.8)

Thus the 95% confidence interval is (-6.5, 12.1)