Question

Please show all work!

3. On the ballot, Illinois voters had the option to select "yes" or "No" when asked if the state’s minimum wage should be increased from $8.25 to $10 an hour by January 1, 2015. Before the actual voting date, a survey with a sample size of 50 found that 61.75% percent of respondents supported the minimum wage.

A) Calculate the 95% and 90% confidence intervals for the minimum wage vote.

B) After the election, the results of vote turned out that 62% percent voters answered “yes”. Do the confidence intervals include the true mean?

C) If the true support rate is 0.6175, what is the sample size should be if we allow 0.05 of margin of error at 95% confidence level?

Answer 1

Given that before the actual voting date , the percentage of voters in the the sample of size 50 in the support of the minimum wage was 61.75 %

So the number of voters in the support of the minimum wage are 31.

The sample proportion is 31/50.

The sample standard deviation is given as

The confidence interval for the population proportion is given as

The z value at 90% confidence interval is 1.68

So , the confidence interval is

The z value at 95% confidence interval is 1.96

So the confidence interval is given as

(b) The true value of the parameter is 0.62.

The confidence interval contains the true value of the parameter.

(c) The z value at the 0.05 level of significance is 1.96

The true calue of the parameter is given 0.6175

which is approximately equal to the sample proportion of voters

Squaring both the sides we see

n = 25