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\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]$\hat{p}=$[/tex] (e.g., 0.123)

Round answers to three decimal places.

Upper bound for [tex]$95\%$[/tex] confidence interval [tex]$=$[/tex] (e.g., 0.122)

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]$\left( \ \_,\ \_ \ \right)$[/tex]

Sagot :

GinnyAnsweridn
To solve this problem, we need to calculate the 95% confidence interval for the estimated proportion (denoted [tex]\(\hat{p}\)[/tex]) of customers that come into the store because of the coffee shop.

Here is the step-by-step solution:

1. Identify the Sample Size and Sample Proportion:
- The sample size ([tex]\(n\)[/tex]) is 75.
- The sample proportion ([tex]\(\hat{p}\)[/tex]) is 0.123.

2. Determine the Z-Score for the Desired Confidence Level:
- For a 95% confidence interval, the Z-score ([tex]\(z^*\)[/tex]) is 1.960. This value is obtained from the standard normal distribution corresponding to a 95% confidence level.

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

4. Calculate the Margin of Error:
- The margin of error (ME) is given by:
[tex]\[ ME = z^* \cdot SE \][/tex]
- Using the values:
[tex]\[ ME = 1.960 \cdot SE \][/tex]

5. Determine the Confidence Interval:
- The lower bound of the confidence interval is:
[tex]\[ \text{Lower bound} = \hat{p} - ME \][/tex]
- The upper bound of the confidence interval is:
[tex]\[ \text{Upper bound} = \hat{p} + ME \][/tex]

6. Calculate the Numerical Values:
- After calculating, we find that the standard error is approximately [tex]\(0.038\)[/tex].
- The margin of error then is approximately [tex]\(0.074\)[/tex].
- Therefore, the confidence interval is:
[tex]\[ \text{Lower bound} = 0.123 - 0.074 = 0.049 \][/tex]
[tex]\[ \text{Upper bound} = 0.123 + 0.074 = 0.197 \][/tex]

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

### Summary
The 95% confidence interval for the proportion of customers that come into the store because of the coffee shop is:
- Upper bound: 0.197
- Lower bound: 0.049

Therefore, you can state that with 95% confidence, the true population proportion is in the interval (0.049, 0.197).
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