Math Problem Statement

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.

Solution

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=p^±zp^(1p^)nCI = \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Where:

  • p^\hat{p} is the sample proportion (in this case, p^=0.49\hat{p} = 0.49),
  • zz is the critical value (which changes depending on the confidence level),
  • nn is the sample size (n=125n = 125).

Step 1: 75% Confidence Interval

For the 75% confidence interval, the critical value z=1.150z = 1.150.

The formula becomes:

CI75=0.49±1.1500.49(10.49)125CI_{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=1.645z = 1.645.

The formula becomes:

CI90=0.49±1.6450.49(10.49)125CI_{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 p^\hat{p}?

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

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Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Proportion Estimation

Formulas

CI = p̂ ± z * sqrt(p̂(1 - p̂) / n)

Theorems

Central Limit Theorem
Normal Approximation for Proportions

Suitable Grade Level

Grades 11-12