Questions: You may need to use the appropriate appendix table or technology to answer this question. The population proportion is 0.26. What is the probability that a sample proportion will be within ± 0.04 of the population proportion for each of the following sample sizes? (your answers to 4 decimal places.) (a) n=100 (b) n=200 (c) n=500

You may need to use the appropriate appendix table or technology to answer this question. The population proportion is 0.26. What is the probability that a sample proportion will be within ± 0.04 of the population proportion for each of the following sample sizes? (your answers to 4 decimal places.) (a) n=100 (b) n=200 (c) n=500

Transcript text: You may need to use the appropriate appendix table or technology to answer this question. The population proportion is 0.26. What is the probability that a sample proportion will be within $\pm 0.04$ of the population proportion for each of the following sample sizes? your answers to 4 decimal places.) (a) $n=100$ $\square$ (b) $n=200$ $\square$ (c) $n=500$ $\square$

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with finding the probability that a sample proportion will be within $ \pm 0.04 $ of the population proportion $ p = 0.26 $ for different sample sizes $ n $.

Step 2: Calculate the Z-scores

The Z-scores for the sample proportions are calculated using the formula:

\[ Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}} \]

where $ \hat{p} $ is the sample proportion, $ p $ is the population proportion, and $ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} $ is the standard deviation of the sample proportion.

For each sample size, we calculate the Z-scores for the bounds:

For $ n = 100 $:
- $ Z_{start} = \frac{0.22 - 0.26}{\sqrt{\frac{0.26 \cdot 0.74}{100}}} \approx -0.9119 $
- $ Z_{end} = \frac{0.30 - 0.26}{\sqrt{\frac{0.26 \cdot 0.74}{100}}} \approx 0.9119 $
For $ n = 200 $:
- $ Z_{start} = \frac{0.22 - 0.26}{\sqrt{\frac{0.26 \cdot 0.74}{200}}} \approx -1.2897 $
- $ Z_{end} = \frac{0.30 - 0.26}{\sqrt{\frac{0.26 \cdot 0.74}{200}}} \approx 1.2897 $
For $ n = 500 $:
- $ Z_{start} = \frac{0.22 - 0.26}{\sqrt{\frac{0.26 \cdot 0.74}{500}}} \approx -2.0391 $
- $ Z_{end} = \frac{0.30 - 0.26}{\sqrt{\frac{0.26 \cdot 0.74}{500}}} \approx 2.0391 $

Step 3: Calculate the Probabilities

Using the Z-scores, we find the probabilities:

For $ n = 100 $: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.9119) - \Phi(-0.9119) \approx 0.6382 \]
For $ n = 200 $: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.2897) - \Phi(-1.2897) \approx 0.8028 \]
For $ n = 500 $: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(2.0391) - \Phi(-2.0391) \approx 0.9586 \]