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 different 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 that corresponds to the desired confidence level.

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

• Formula for confidence interval (mean, known σ):

$\bar{x} \pm Z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$

Step-by-Step Guidance

1. Recall that the critical Z value corresponds to the area in the tails of the standard normal distribution that is not included in the confidence level. For a two-tailed interval, the area in each tail is $\alpha/2$ where $\alpha = 1 - \text{confidence level}$.

2. For each confidence level, calculate $\alpha$ and then $\alpha/2$.

3. Use a standard normal (Z) table or calculator to find the Z value that leaves $\alpha/2$ in the upper tail (or equivalently, has cumulative area $1 - \alpha/2$ to the left).

4. Write down the Z value for each confidence level, but do not compute the final values yet.

Try solving on your own before revealing the answer!

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 – Interpretation and Properties

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

• Critical Value: The Z or t value associated with the confidence level.

Step-by-Step Guidance

1. Recall the formula for the confidence interval: $\bar{x} \pm Z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$.

2. Notice that the only part of the formula affected by the confidence level is the critical value $Z_{\alpha/2}$.

3. As the confidence level increases, $\alpha$ decreases, so $Z_{\alpha/2}$ increases.

4. Think about how increasing $Z_{\alpha/2}$ affects the width of the interval.

Try solving on your own before revealing the answer!

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 in a confidence interval for the mean.

Key Terms and Formula

• Margin of Error (E): The amount added and subtracted from the sample mean to create the confidence interval.

• Formula:

$E = Z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$

• Sample mean ($\bar{x}$): The center of the interval, not the width.

• Sample size ($n$): Appears in the denominator.

• Confidence level: Determines $Z_{\alpha/2}$.

• Standard deviation ($\sigma$): Appears in the numerator.

• Median, mode: Measures of center, not used in the margin of error formula.

Step-by-Step Guidance

1. Write out the formula for the margin of error.

2. Identify which variables in the formula correspond to the options given.

3. Consider whether the sample mean, median, or mode appear in the formula.

4. Determine which factors directly affect the margin of error.

Try solving on your own before revealing the answer!

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 Population Proportion

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

Key Terms and Formula

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

• Confidence interval for a proportion:

$\hat{p} \pm Z_{\alpha/2} \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$

• $x$ = number of successes (heads)

• $n$ = total trials (flips)

• $Z_{\alpha/2}$ = critical value for 90% confidence

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.

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. Add and subtract the margin of error from $\hat{p}$ to get the interval (stop before calculating the final numbers).

Try solving on your own before revealing the answer!

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 Population Proportion

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

Key Terms and Formula

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

• Confidence interval for a proportion:

$\hat{p} \pm Z_{\alpha/2} \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$

• $n = 75$

• $x = 60$

• $Z_{\alpha/2}$ for 95% confidence

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. Add and subtract the margin of error from $\hat{p}$ to get the interval (stop before calculating the final numbers).

Try solving on your own before revealing the answer!

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 mean when the population standard deviation is known.

Key Terms and Formula

• Sample mean ($\bar{x}$): 77 ounces

• Population standard deviation ($\sigma$): 7 ounces

• Sample size ($n$): 36

• Confidence interval formula:

$\bar{x} \pm Z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$

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. Add and subtract the margin of error from the sample mean to get the interval (stop before calculating the final numbers).

Try solving on your own before revealing the answer!