Step-by-Step Guidance for STAT 101 Assignment #6 (Probability & Confidence Intervals)

Q1. What is the probability that more than 40 employees in a random sample of 400 participate in the professional development program, given that historically 14% participate?

Background

Topic: Sampling Distribution of the Sample Proportion (Normal Approximation to the Binomial)

This question tests your understanding of how to use the normal approximation to the binomial distribution to find probabilities about sample proportions.

Key Terms and Formulas

• Population proportion: $p$

• Sample size: $n$

• Sample proportion: $\hat{p}$

• Mean of $\hat{p}$: $\mu_{\hat{p}} = p$

• Standard deviation of $\hat{p}$: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Normal approximation: Use when $np \geq 10$ and $n(1-p) \geq 10$

• Convert to $z$-score: $z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$

Step-by-Step Guidance

1. Identify the known values: $p = 0.14$, $n = 400$.

2. Calculate the sample proportion corresponding to 40 employees: $\hat{p} = \frac{40}{400}$.

3. Compute the mean and standard deviation of the sampling distribution: $\mu_{\hat{p}} = p$, $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.

4. Convert the sample proportion to a $z$-score using the formula: $z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$.

5. Use the standard normal table to find the probability that $z$ is greater than the value you calculated.

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Q2. What is the probability that between 18% and 25% of a random sample of 250 clients experience positive returns, given that 20% of all clients do?

Background

Topic: Sampling Distribution of the Sample Proportion (Normal Approximation to the Binomial)

This question tests your ability to find the probability that a sample proportion falls within a given range, using the normal approximation.

Key Terms and Formulas

• Population proportion: $p = 0.20$

• Sample size: $n = 250$

• Sample proportion: $\hat{p}$

• Standard deviation: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Convert to $z$-scores: $z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$

Step-by-Step Guidance

1. Identify the lower and upper bounds for the sample proportion: $\hat{p}_1 = 0.18$, $\hat{p}_2 = 0.25$.

2. Calculate the mean and standard deviation of the sampling distribution: $\mu_{\hat{p}} = p$, $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.

3. Convert both bounds to $z$-scores: $z_1 = \frac{0.18 - 0.20}{\sigma_{\hat{p}}}$ and $z_2 = \frac{0.25 - 0.20}{\sigma_{\hat{p}}}$.

4. Use the standard normal table to find the probabilities corresponding to $z_1$ and $z_2$.

5. Subtract the lower probability from the upper probability to find the probability that $\hat{p}$ is between 0.18 and 0.25.

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Q3. Construct and interpret a 95% confidence interval for the true proportion of all trained employees who would avoid clicking phishing links, based on a sample of 200 with 160 successes.

Background

Topic: Confidence Interval for a Population Proportion

This question tests your ability to construct and interpret a confidence interval for a population proportion using sample data.

Key Terms and Formulas

• Sample proportion: $\hat{p} = \frac{x}{n}$

• Standard error: $SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Critical value for 95% confidence: $z^* \approx 1.96$

• Confidence interval: $\hat{p} \pm z^* \times SE$

Step-by-Step Guidance

1. Calculate the sample proportion: $\hat{p} = \frac{160}{200}$.

2. Compute the standard error: $SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{200}}$.

3. Identify the critical value for a 95% confidence interval ($z^* = 1.96$).

4. Set up the confidence interval formula: $\hat{p} \pm z^* \times SE$.

5. Calculate the lower and upper bounds of the interval using the values from the previous steps.

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Q4. What is the minimum sample size required to estimate the proportion of satisfied customers within a margin of error of 5% at 90% confidence, given an estimated proportion of 75%?

Background

Topic: Sample Size Determination for Estimating a Proportion

This question tests your ability to determine the minimum sample size needed to achieve a specified margin of error for a confidence interval for a proportion.

Key Terms and Formulas

• Estimated proportion: $\hat{p} = 0.75$

• Margin of error: $E = 0.05$

• Critical value for 90% confidence: $z^* \approx 1.645$

• Sample size formula: $n = \frac{(z^*)^2 \hat{p}(1-\hat{p})}{E^2}$

Step-by-Step Guidance

1. Identify the estimated proportion ($\hat{p} = 0.75$), margin of error ($E = 0.05$), and critical value ($z^* = 1.645$ for 90% confidence).

2. Plug these values into the sample size formula: $n = \frac{(z^*)^2 \hat{p}(1-\hat{p})}{E^2}$.

3. Calculate the numerator: $(z^*)^2 \hat{p}(1-\hat{p})$.

4. Calculate the denominator: $E^2$.

5. Divide the numerator by the denominator to find the minimum sample size (always round up to the next whole number).

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Q5. Construct and interpret a 99% confidence interval for the difference in return rates between Nike and Adidas shoes, based on two independent samples (Nike: 150/200, Adidas: 120/180).

Background

Topic: Confidence Interval for the Difference Between Two Proportions

This question tests your ability to construct and interpret a confidence interval for the difference between two independent population proportions.

Key Terms and Formulas

• Sample proportions: $\hat{p}_1 = \frac{150}{200}$ (Nike), $\hat{p}_2 = \frac{120}{180}$ (Adidas)

• Standard error: $SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Critical value for 99% confidence: $z^* \approx 2.576$

• Confidence interval: $(\hat{p}_1 - \hat{p}_2) \pm z^* \times SE$

Step-by-Step Guidance

1. Calculate the sample proportions for each group: $\hat{p}_1 = \frac{150}{200}$, $\hat{p}_2 = \frac{120}{180}$.

2. Compute the standard error: $SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{200} + \frac{\hat{p}_2(1-\hat{p}_2)}{180}}$.

3. Identify the critical value for a 99% confidence interval ($z^* = 2.576$).

4. Set up the confidence interval formula: $(\hat{p}_1 - \hat{p}_2) \pm z^* \times SE$.

5. Calculate the lower and upper bounds of the interval using the values from the previous steps.

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