Question

1. In a USA Today/Gallup poll, 768 of 1024 randomly selected adult Americans stated that a candidate’s positions on the issue of family values are extremely or very important in determining their vote for president.
2. Obtain a point estimate for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president.
3. Verify that the requirements for constructing a confidence interval for p are satisfied
4. Construct 90%, 95% and 99% confidence intervals for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president. Be sure to give a proper interpretation of each interval
5. What do you notice about the width of each confidence interval as the confidence level increases?

Answer 1

n = number of randomly selected adult Americans= 1024

x = number of adults who stated that a candidate’s positions on the issue of family values are extremely or very important in determining their vote for president = 768

Sample proportion:

Point estimate for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is 0.75.

For Confidence level = c = 0.90

90% confidence intervals for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is

where zc is z critical value for (1+c)/2 = (1+0.90)/2 = 0.95 is

zc = 1.645 (From statistical table of z values, average of 1.64 and 1.65, (1.64+1.65)/2 = 1.645)

(Round to 4 decimal)

90% confidence interval for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is (0.7277, 0.7723)

Interpretation: We are 90% confident that true proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president will lie between the interval (0.7277, 0.7723)

For Confidence level = c = 0.95

95% confidence intervals for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is

where zc is z critical value for (1+c)/2 = (1+0.95)/2 = 0.975 is

zc = 1.96 (From statistical table of z values)

(Round to 4 decimal)

95% confidence interval for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is (0.7235, 0.7765)

Interpretation: We are 95% confident that true proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president will lie between the interval (0.7235, 0.7765)

For Confidence level = c = 0.99

99% confidence intervals for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is

where zc is z critical value for (1+c)/2 = (1+0.99)/2 = 0.995 is

zc = 2.58 (From statistical table of z values)

(Round to 4 decimal)

99% confidence interval for the proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president is (0.7151, 0.7849)

Interpretation: We are 99% confident that true proportion of adult Americans for which the issue of family values is extremely or very important in determining their vote for president will lie between the interval (0.7151, 0.7849)

90% Confidence interval is (0.7277, 0.7723)

Width = Upper limit - lower limit

= 0.7723 - 0.7277

= 0.0446

Width of 90% confidence interva is 0.0446

95% Confidence interval is (0.7235, 0.7765)

Width = Upper limit - lower limit

= 0.7765 - 0.7235

= 0.053

Width of 95% confidence interva is 0.053

99% Confidence interval is (0.7151, 0.7849)

Width = Upper limit - lower limit

= 0.7849 - 0.7151

= 0.0698

Width of 99% confidence interva is 0.0698

We can say that as confidence level increases width of confidence interval also increases.