\begin{tabular}{|l|l|}
\hline
0.90 & [tex]$z^\ \textless \ em\ \textgreater \ =1.645$[/tex] \\
\hline
0.95 & [tex]$z^\ \textless \ /em\ \textgreater \ =1.960$[/tex] \\
\hline
0.99 & [tex]$z^*=2.576$[/tex] \\
\hline
\end{tabular}

The manager of a bookstore with a coffee shop wants to know the proportion of customers that come into the store because of the coffee shop. A random sample of 75 customers was polled. Use Sheet 1 of the Excel file linked above to calculate [tex]$\hat{p}$[/tex] and the [tex]$95\%$[/tex] confidence interval.

[tex]$\hat{p}=\operatorname{Ex}: 0.123$[/tex]

Round answers to three decimal places.

Upper bound for [tex]$95\%$[/tex] confidence interval [tex]$=$[/tex] Ex: 0.123

Lower bound for [tex]$95\%$[/tex] confidence interval [tex]$=$[/tex] [tex]$\square$[/tex]

The manager can say with [tex]$95\%$[/tex] confidence that the true population proportion of the customers that come into the store because of the coffee shop is in the interval: ([tex]$\square$[/tex], [tex]$\square$[/tex])



Answer :

To find the 95% confidence interval for the proportion of customers who visit the bookstore because of the coffee shop, follow these steps:

1. Identify the Sample Proportion [tex]\(\hat{p}\)[/tex]:
The sample proportion, [tex]\(\hat{p}\)[/tex], is given as 0.123 from the random sample of 75 customers.

2. Determine the Sample Size [tex]\(n\)[/tex]:
The sample size is given as [tex]\(n = 75\)[/tex].

3. Identify the Z-Score for the Desired Confidence Level:
For a 95% confidence level, the corresponding Z-score ([tex]\(z^*\)[/tex]) is 1.960.

4. Calculate the Standard Error (SE):
The standard error is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} (1-\hat{p})}{n}} \][/tex]
Plugging in the given values:
[tex]\[ SE = \sqrt{\frac{0.123 \cdot (1 - 0.123)}{75}} = 0.038 \][/tex]

5. Calculate the Margin of Error (MoE):
The margin of error is found by multiplying the Z-score by the standard error:
[tex]\[ MoE = z^* \cdot SE = 1.960 \cdot 0.038 = 0.074 \][/tex]

6. Calculate the Lower and Upper Bounds of the Confidence Interval:
- Lower Bound:
[tex]\[ \text{Lower Bound} = \hat{p} - MoE = 0.123 - 0.074 = 0.049 \][/tex]
- Upper Bound:
[tex]\[ \text{Upper Bound} = \hat{p} + MoE = 0.123 + 0.074 = 0.197 \][/tex]

7. Conclusion:
The manager can say with 95% confidence that the true population proportion of customers who come into the store because of the coffee shop is in the interval:
[tex]\[ (0.049, 0.197) \][/tex]

Rounded to three decimal places, the lower bound is 0.049 and the upper bound is 0.197.