Comprehensive Step-by-Step Guidance for College Statistics Final Exam Study Guide

Q1. What is the critical Z value used to create the following confidence intervals?

• a) 80%

• b) 90%

• c) 95%

• d) 99%

Background

Topic: Confidence Intervals and the Standard Normal (Z) Distribution

This question tests your understanding of how to find the critical Z values (also called Z* or Zα/2) for various confidence levels, which are used in constructing confidence intervals for population parameters when the population standard deviation is known.

Key Terms and Formulas

• Critical Z value (Zα/2): The value from the standard normal distribution such that the area between -Zα/2 and Zα/2 equals the desired confidence level.

• Confidence Level (CL): The probability that the interval estimate contains the true population parameter.

• α (alpha): The probability that the interval does not contain the parameter (α = 1 - CL).

Step-by-Step Guidance

1. For each confidence level, calculate α by subtracting the confidence level from 1. For example, for a 95% confidence level, α = 1 - 0.95 = 0.05.

2. Divide α by 2 to find the area in each tail of the normal distribution: $\alpha/2$.

3. Use the standard normal (Z) table to find the Z value such that the area to the left of Z is $1 - \alpha/2$.

4. Repeat this process for each confidence level (80%, 90%, 95%, 99%).

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Q2. Assuming all else is equal, what is the key impact confidence level has on the width of a confidence interval?

Background

Topic: Confidence Intervals

This question is about understanding how changing the confidence level affects the width of a confidence interval, assuming other factors (like sample size and standard deviation) remain constant.

Key Terms

• Confidence Level: The probability that the interval contains the true parameter.

• Width of Confidence Interval: The range between the lower and upper bounds of the interval.

Step-by-Step Guidance

1. Recall that the width of a confidence interval depends on the critical value (Z or t), the standard deviation, and the sample size.

2. As the confidence level increases, the critical value (Z or t) also increases.

3. Consider how a larger critical value affects the margin of error and, consequently, the width of the interval.

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Q3. Which of the following impact the margin of error of a confidence interval about the mean: Sample mean, sample size, confidence level, standard deviation, median, mode?

Background

Topic: Margin of Error in Confidence Intervals

This question tests your understanding of which factors influence the margin of error when constructing a confidence interval for the mean.

Key Terms and Formulas

• Margin of Error (E): $E = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$ (for known population standard deviation)

• Sample Mean (\bar{x}): The average of the sample data.

• Sample Size (n): The number of observations in the sample.

• Standard Deviation (\sigma): A measure of variability in the population.

• Confidence Level: Determines the critical value (Z or t).

• Median and Mode: Other measures of central tendency.

Step-by-Step Guidance

1. Review the formula for the margin of error and identify which variables are present.

2. Consider whether the sample mean, median, or mode appear in the formula for margin of error.

3. Think about how changing the sample size, confidence level, or standard deviation would affect the margin of error.

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Q4. Compute the 90% confidence interval for the proportion of heads observed if we flip a coin 100 times and observe 65 heads show up.

Background

Topic: Confidence Interval for a Proportion

This question asks you to construct a confidence interval for a population proportion based on sample data.

Key Terms and Formulas

• Sample Proportion (\hat{p}): $\hat{p} = \frac{x}{n}$

• Confidence Interval Formula: $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• n: Sample size

• x: Number of successes (heads)

• Zα/2: Critical value for the desired confidence level

Step-by-Step Guidance

1. Calculate the sample proportion: $\hat{p} = \frac{65}{100}$.

2. Find the critical Z value for a 90% confidence interval (use your answer from Q1).

3. Compute the standard error: $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

4. Multiply the critical Z value by the standard error to get the margin of error.

5. Set up the confidence interval: $\hat{p} \pm$ margin of error.

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Q5. 75 people are surveyed to see if they enjoy the new park created by the town. Of these 75 people, 60 stated they enjoyed the new park. Compute the 95% confidence interval for the proportion of people who enjoyed the new park.

Background

Topic: Confidence Interval for a Proportion

This question is similar to Q4, but with different sample data and a different confidence level.

Key Terms and Formulas

• Sample Proportion (\hat{p}): $\hat{p} = \frac{60}{75}$

• Confidence Interval Formula: $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• n: Sample size

• x: Number of successes

• Zα/2: Critical value for the desired confidence level

Step-by-Step Guidance

1. Calculate the sample proportion: $\hat{p} = \frac{60}{75}$.

2. Find the critical Z value for a 95% confidence interval.

3. Compute the standard error: $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

4. Multiply the critical Z value by the standard error to get the margin of error.

5. Set up the confidence interval: $\hat{p} \pm$ margin of error.

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Q6. A popular breakfast company fills their cereal bags with an average of 80 ounces of cereal and a standard deviation of 7 ounces. We sample 36 bags and find the average weight of those cereal bags to be 77 ounces. Compute the 95% confidence interval for the mean based on our sample.

Background

Topic: Confidence Interval for the Mean (Known Population Standard Deviation)

This question asks you to construct a confidence interval for the population mean using the sample mean, sample size, and known population standard deviation.

Key Terms and Formulas

• Sample Mean (\bar{x}): 77 ounces

• Population Standard Deviation (\sigma): 7 ounces

• Sample Size (n): 36

• Confidence Interval Formula: $\bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Step-by-Step Guidance

1. Identify the sample mean, population standard deviation, and sample size.

2. Find the critical Z value for a 95% confidence interval.

3. Calculate the standard error: $\frac{\sigma}{\sqrt{n}}$.

4. Multiply the critical Z value by the standard error to get the margin of error.

5. Set up the confidence interval: $\bar{x} \pm$ margin of error.

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Q7. A new process has been created to help reduce the recovery time of a certain medical procedure. 100 people who underwent this procedure had a mean recovery time of 46.3 days. If the population standard deviation can be assumed to be 12 days, compute the 90% confidence interval for the mean recovery time.

Background

Topic: Confidence Interval for the Mean (Known Population Standard Deviation)

This question is similar to Q6, but with different sample data and a different confidence level.

Key Terms and Formulas

• Sample Mean (\bar{x}): 46.3 days

• Population Standard Deviation (\sigma): 12 days

• Sample Size (n): 100

• Confidence Interval Formula: $\bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

Step-by-Step Guidance

1. Identify the sample mean, population standard deviation, and sample size.

2. Find the critical Z value for a 90% confidence interval.

3. Calculate the standard error: $\frac{\sigma}{\sqrt{n}}$.

4. Multiply the critical Z value by the standard error to get the margin of error.

5. Set up the confidence interval: $\bar{x} \pm$ margin of error.

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