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 90% confidence interval for the true difference between the population means. Assume that the populations of voters' performance ratings are normally distributed for both this year and last year.

Rating (last year)	59	48	56	56	86	64	50	72	81	66
Rating (this year)	77	57	54	60	84	93	65	60	87	77

Step 3 of 4 :  

Calculate the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.

Step 4 of 4 :

Construct the 90% confidence interval. Round your answer to one decimal place.

Answer 1

As the data refers to the the legislator's performance from the same ten randomly selected voters for last year and this year, Paired(dependent) confidence interval for the true difference between the population means is appropriate.

xi: Rating (last year) from the ith voter of the sample

yi: Rating (this year) from the ith voter of the sample

di :(This year rating - last year rating ) : (yi-xi) :difference in the rating from the ith voter

Sample size : n=10

: Sample mean difference

sd : Sample standard deviation of the difference

Margin of error formula

formula for confidence interval for the true difference between the population means(Paired)

for 90% confidence level = (100-90)/100 =0.10 ; /2 =0.10/2=0.05; n-1=10-1=9

t/2,n-1 = t0.05 for 9 degrees of freedom : t0.05,9 = 1.83311293

x: Rating (last year)	y: Rating (this year)	d:This year -Last year
59	77	18	10.4	108.16
48	57	9	1.4	1.96
56	54	-2	-9.6	92.16
56	60	4	-3.6	12.96
86	84	-2	-9.6	92.16
64	93	29	21.4	457.96
50	65	15	7.4	54.76
72	60	-12	-19.6	384.16
81	87	6	-1.6	2.56
66	77	11	3.4	11.56
		=76		=1218.4
		=7.6

Step 3 of 4 :

Margin of error formula

Calculate the margin of error to be used in constructing the confidence interval.

Margin of error = 6.744704

Step 4 of 4

90% confidence interval for the true difference between the population means

90% confidence interval for the true difference between the population means = (0.855296 ,14.344704 )

90% confidence interval for the true difference between the population means = (0.9 ,14.3 )