Question

Based on a sample data of 64 children mean age of children in the summer camp A is 12,4 years with Standard deviation 3 years. Estimate the average age of children in whole population. Population size is 8100

a) Find the point estimation for average age
b) With 95% confidence, what is the margin of error? With 90% confidence, what is the margin of error?
c) What is the 95% confidence interval estimate of the population mean (mean children age in all summer camps)?
d) What would happen to the confidence interval if the confidence level would be increased to 99%? Would the confidence interval become wider or narrower? Why? Assume that the sample size remains the same.
e) Do you agree or disagree with the claim that the used sample is too small to guarantee maximum tolerable error (acceptable margin of error) +-2 yers? Why?

Answer 1

The provided information is,

sample mean==12.4 years

sample standard deviation=s=3 years

sample size=n=64

Population size=N=8100

(a)

Since, the sample mean is the unbiased estimate of the population mean, the point estimate of mean age of children will be same as the mean age of the sample. That is, the estimated value of mean age is 12.4 years.

Therefore, the estimated value of mean age is 12.4 years.

(b)

The margin of error can be computed using the formula,

Here, is the tabulated value of t distribution at (n-1) degrees of freedom and significance level.

At 95% confidence level, that is, 5% significance level, the value of   is obtained using Excel function. The screenshot is shown below.

Therefore, the margin of error is calculated as follows:

At 90% confidence level, that is, 10% significance level, the value of   is obtained using Excel function. The screenshot is shown below.

Therefore, the margin of error is calculated as follows:

Hence, the 95% and 90% margin of error are 0.7493 and 0.6259, respectively.

(c)

The 95% confidence interval for population mean is calculated as follows:

Hence, the 95% confidence interval is (11.6507,13.1493).

(d)

The confidence level is directly proportional with the confidence level. So, the width of the confidence interval will increase due to increase in the confidence level.

In the question, the confidence level is increased from 95% to 99%. Hence, the width of the confidence interval will increased.

Therefore, it can be said that the confidence interval will be wider when the confidence level is increased from 95% to 99%.

(e)

According to the question, the maximum value of the margin of error is 2 years.

In the provided scenario, the calculated margin of error at 95% confidence level with a sample of size 64 is 0.7493. So, it can be said that the sample size is not too small to meet the maximum tolerable error.