Question

Sarah Anderson is running for mayor in a small rural town and would like to estimate the age of women voters. The town has 636 registered women voters on the records. Sarah found that a random sample of 70 women voters had an average age of 42.7 years and a standard deviation of 7.5 years. Determine the 90% confidence interval for the average of women voters in this town.

Answer 1

( 41.23, 44.17) years

Therefore at 90% confidence interval= ( 41.23, 44.17) years

Explanation:

Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.

The confidence interval of a statistical data can be written as.

x+/-zr/√n

Given that;

Mean x = 42.7

Standard deviation r = 7.5

Number of samples n = 70

Confidence interval = 90%

z(at 90% confidence) = 1.645

Substituting the values we have;

42.7+/-1.645(7.5/√70)

42.7+/-1.645(0.8964)

42.7+/-1.47

= ( 41.23, 44.17)

Therefore at 90% confidence interval= ( 41.23, 44.17) years

Answer 2

The 90% confidence interval for the average of women voters in this town is between 30.193 years and 55.207 years

Explanation:

We have the standard deviation for the sample, so we use the t-distribution to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 70 - 1 = 69

90% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 69 degrees of freedom(y-axis) and a confidence level of
`1 - (1 - 0.9)/(2) = 0.95`. So we have T = 1.6676

The margin of error is:

M = T*s = 1.6676*7.5 = 12.507

In which s is the standard deviation of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 42.7 - 12.507 = 30.193 years

The upper end of the interval is the sample mean added to M. So it is 42.7 + 12.507 = 55.207 years

The 90% confidence interval for the average of women voters in this town is between 30.193 years and 55.207 years