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=0.55 . You have taken a random sample of size n=125 from the population and found that the proportion of the sample  that has the characteristic is sample proportion=0.49. Your sample is Sample 1 in the table.  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.

To calculate the confidence intervals for the population proportion based on the sample proportion, we will use the formula for a confidence interval (CI) for a population proportion:

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

Where:
- \(\hat{p}\) is the sample proportion (in this case, \(\hat{p} = 0.49\)),
- \(z\) is the critical value (which changes depending on the confidence level),
- \(n\) is the sample size (\(n = 125\)).

### Step 1: 75% Confidence Interval
For the 75% confidence interval, the critical value \(z = 1.150\).

The formula becomes:

\[
CI_{75} = 0.49 \pm 1.150 \cdot \sqrt{\frac{0.49 \cdot (1 - 0.49)}{125}}
\]

Let's compute this.

### Step 2: 90% Confidence Interval
For the 90% confidence interval, the critical value \(z = 1.645\).

The formula becomes:

\[
CI_{90} = 0.49 \pm 1.645 \cdot \sqrt{\frac{0.49 \cdot (1 - 0.49)}{125}}
\]

Let me calculate both of these confidence intervals.The confidence intervals for the population proportion based on the sample are as follows:

1. **75% Confidence Interval**: 
   - Lower bound: 0.439
   - Upper bound: 0.541

2. **90% Confidence Interval**: 
   - Lower bound: 0.416
   - Upper bound: 0.564

These intervals provide a range of values within which the true population proportion is likely to lie with the given levels of confidence.

Would you like to see these intervals plotted on a graph or need further details?

Here are 5 related questions to consider:
1. How do confidence intervals change with larger sample sizes?
2. What is the effect of a higher confidence level on the width of the interval?
3. How is the critical value for a confidence interval determined?
4. What are the assumptions behind using a normal approximation for confidence intervals of proportions?
5. How does the standard error of the sample proportion change with different values of \(\hat{p}\)?

**Tip**: Higher confidence levels result in wider intervals to ensure greater certainty that the population parameter lies within the range.

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=0.55 . You have taken a random sample of size n=125 from the population and found that the proportion of the sample that has the characteristic is sample proportion=0.49. Your sample is Sample 1 in the table. 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.

Learn how to calculate and graph 75% and 90% confidence intervals for a population proportion with sample data. This example involves using the normal approximation for proportions, with key critical values and sample size data. Ideal for students in Grades 11-12.

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