# A questionnaire on the issue of substance abuse problems was mailed to all XYZ city businesses....

## Question:

A questionnaire on the issue of substance abuse problems was mailed to all XYZ city businesses. Of the 72 businesses that responded to the survey, 54 admitted that they had employees whose performance was affected by substance abuse.

(a) Using the sample data, compute a 95% confidence interval for the proportion of all XYZ city businesses with substance problems.

(b) Explain what is the margin of error.

(c) Interpret the confidence interval in (a).

## Confidence Interval:

The confidence interval gives the lower and the upper limits for the population proportion on the basis of the sample proportion. The sample proportion is also used to estimate the population proportion. The sample proportion is an unbiased estimate of the population proportion.

Given that,

Sample size, {eq}n = 72 {/eq}

Sample proportion, {eq}\hat{p} = \dfrac{54}{72} = 0.75 {/eq}

a)

The 95% confidence coefficient for the proportion is defined as:

{eq}\hat{p} \pm z_{0.05/2}\times \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} {/eq}

Excel function for the confidence coefficient:

=NORMINV(0.05/2,0,1)

{eq}0.75 \pm 1.96\times \sqrt{\dfrac{0.75(1- 0.75)}{72}}\\ (0.6499, \ 0.850) {/eq}

b)

The margin of error is calculated as:

{eq}ME = \dfrac{0.85 - 0.6499}{2}\\ ME = 0.10005 {/eq}

The confidence interval will provide the lower and the upper limits for the population parameter. The true value of the population proportion will lie within the confidence interval.