Question

A poll reported that only 580 out of a total of 1649 adults in a particular region said they had a​ "great deal of​ confidence" or​ "quite a lot of​ confidence" in the public school system. This was down 5 percentage points from the previous year. Assume the conditions for using the CLT are met. Complete parts​ (a) through​ (d) below.

1. Find a 95% confidence interval for the proportion that express a great deal of confidence or quite a lot of confidence in the public​ schools, and interpret this interval. (____,____) (3 decimels

2. We are Answer ____% confident that the population proportion of adults having a great deal or quite a lot of confidence in the public schools is between Answer ___ and ____.

3. Find an  80% confidence interval and interpret it. The 80% confidence interval for the proportion that express a great deal of confidence or quite a lot of confidence in the public schools is Answer  (____,____) (Round 3 decimal places)

4. Find the width of each interval by subtracting the lower proportion from the upper​ proportion, and state which interval is wider.The width of the 95% confidence interval is (Answer) ____ and the width of the 80% confidence interval is (Answer) ____ The 95% interval is wider. ​(Round to three decimal places as​ needed.)

Answer 1

1)

x= 580, n= 1649, c=  95%

formula for confidence interval is

where Zc is the z critical value for c= 95%

Zc= 1.96

0.32868 < P < 0.37478

Answer = ( 0.329 , 0.375)

2)

We are 95 % confident that the population proportion of adults having a great deal or quite a lot of confidence in the public schools is between 0.329   and  0.375

3)

x= 580, n= 1649, c= 80%

formula for confidence interval is

where Zc is the z critical value for c= 80%

Zc= 1.28

0.33666 < P < 0.3668

Answer = ( 0.337 , 0.367)

The 80% confidence interval for the proportion that express a great deal of confidence or quite a lot of confidence in the public schools is (0.337 , 0.367 )

4)

for C= 95%

width = Upper proportion - Lower proportion

width = 0.375 - 0.329

Width = 0.046

for C= 80%

width = Upper proportion - Lower proportion

width = 0.367 -   0.337

Width = 0.03

The width of the 95% confidence interval is 0.046  and the width of the 80% confidence interval is 0.03 The 95% interval is wider.