You are looking at a population and are interested in the proportion p

 that has a certain characteristic. Unknown to you, this population proportion is =

p

0.45

. You have taken a random sample of size =

n

85

 from the population and found that the proportion of the sample that has the characteristic is =

p

0.37

. Your sample is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.)



(a)Based on Sample 1, graph the 75

%

 and 90

%

 confidence intervals for the population proportion. Use 1.150

 for the critical value for the 75

%

 confidence interval, and use 1.645

 for the critical value for the 90

%

 confidence interval. (If necessary, consult a list of formulas.)Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with two decimal places.

For the points ( and ), enter the population proportion, 0.45

.

75% confidence interval

0.25

0.63













 90% confidence interval

0.25

0.63













To compute the confidence intervals for the population proportion based on Sample 1, we'll use the following formula for the confidence interval of a population proportion:

\[
\text{CI} = \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \(\hat{p}\) is the sample proportion (0.37 in this case),
- \(n\) is the sample size (85 in this case),
- \(z\) is the critical value (provided for 75% and 90% confidence levels),
- \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) is the standard error of the sample proportion.

### Step 1: Calculate the Standard Error (SE)
The standard error (SE) is calculated as:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.37(1 - 0.37)}{85}} = \sqrt{\frac{0.37 \times 0.63}{85}} \approx \sqrt{\frac{0.2331}{85}} \approx \sqrt{0.002742} \approx 0.05237
\]

### Step 2: Calculate the 75% Confidence Interval
For the 75% confidence interval, the critical value \(z = 1.150\).

The confidence interval formula is:
\[
CI = 0.37 \pm 1.150 \times 0.05237
\]
Calculate the margin of error:
\[
1.150 \times 0.05237 \approx 0.06023
\]

Now calculate the lower and upper bounds:
- Lower bound: \(0.37 - 0.06023 = 0.30977 \approx 0.31\)
- Upper bound: \(0.37 + 0.06023 = 0.43023 \approx 0.43\)

Thus, the **75% confidence interval** is \((0.31, 0.43)\).

### Step 3: Calculate the 90% Confidence Interval
For the 90% confidence interval, the critical value \(z = 1.645\).

The confidence interval formula is:
\[
CI = 0.37 \pm 1.645 \times 0.05237
\]
Calculate the margin of error:
\[
1.645 \times 0.05237 \approx 0.08614
\]

Now calculate the lower and upper bounds:
- Lower bound: \(0.37 - 0.08614 = 0.28386 \approx 0.28\)
- Upper bound: \(0.37 + 0.08614 = 0.45614 \approx 0.46\)

Thus, the **90% confidence interval** is \((0.28, 0.46)\).

### Summary of Results:
- **75% Confidence Interval**: (0.31, 0.43)
- **90% Confidence Interval**: (0.28, 0.46)

If plotted on a graph, you'd mark the population proportion \(p = 0.45\) alongside these intervals.

Would you like more details on any of these calculations, or have any questions?

Here are five questions to further explore this topic:
1. How does increasing the sample size affect the width of the confidence intervals?
2. Why is the 90% confidence interval wider than the 75% confidence interval?
3. What would happen to the intervals if the sample proportion was closer to 0.45?
4. How is the choice of confidence level related to the margin of error?
5. What are the implications if the population proportion falls outside of the calculated confidence intervals?

**Tip:** The width of a confidence interval decreases as the sample size increases, which means more precise estimates for larger samples.

You are looking at a population and are interested in the proportion p that has a certain characteristic. Unknown to you, this population proportion is p = 0.45. You have taken a random sample of size n = 85 from the population and found that the proportion of the sample that has the characteristic is p = 0.37. Based on Sample 1, graph the 75% and 90% confidence intervals for the population proportion. Use 1.150 for the critical value for the 75% confidence interval, and use 1.645 for the 90% confidence interval.

Learn how to calculate the 75% and 90% confidence intervals for a population proportion with a sample of size 85, sample proportion of 0.37, and given critical values. This step-by-step solution covers the standard error calculation and confidence interval formula. Suitable for undergraduate statistics students.

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