Q1. A sample of 36 yields a mean weight of 1.02 pounds per box. Construct an 80% confidence interval for the mean weight of all boxes, assuming the weight is normally distributed with a population standard deviation of 0.20 pounds. (2 decimal places)

Background

Topic: Confidence Intervals for the Mean (Normal Distribution, Known Population Standard Deviation)

This question tests your ability to construct a confidence interval for the population mean when the population standard deviation is known and the sample size is given.

Key Terms and Formulas:

• Confidence Interval (CI): Range of values likely to contain the population mean.

• Standard Error (SE):

• Z-score for confidence level: Find for 80% confidence.

• CI formula:

Step-by-Step Guidance

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

2. Calculate the standard error:

3. Determine the z-score for an 80% confidence interval. For 80%, , so . Find .

4. Set up the confidence interval formula:

Try solving on your own before revealing the answer!

Q2. A car dealership wants to estimate the mean mpg of its new model car with 80% confidence. The population is normally distributed, and a sample of 25 cars yields a mean of 96.52 mpg and a population standard deviation of 2.82. Construct an 80% confidence interval for the population mean mpg.

Background

Topic: Confidence Intervals for the Mean (Normal Distribution, Known Population Standard Deviation)

This question tests your ability to construct a confidence interval for the mean mpg using a sample mean, known population standard deviation, and sample size.

Key Terms and Formulas:

• Sample mean (), population standard deviation (), sample size ()

• Standard Error:

• Z-score for 80% confidence:

• Confidence Interval:

Step-by-Step Guidance

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

2. Calculate the standard error:

3. Determine the z-score for an 80% confidence interval ().

4. Set up the confidence interval formula:

Try solving on your own before revealing the answer!

Q3. A random sample of size 36 is selected from a normally distributed population with a mean of 20.0 and a population standard deviation of 4.0. What is the probability that the sample mean will be between 19.0 and 21.0?

Background

Topic: Sampling Distribution of the Mean (Normal Distribution)

This question tests your ability to use the sampling distribution of the mean to find the probability that the sample mean falls within a specified range.

Key Terms and Formulas:

• Sampling distribution mean:

• Standard error:

• Z-score:

• Probability: Use the standard normal table to find

Step-by-Step Guidance

1. Calculate the standard error:

2. Find the z-scores for 19.0 and 21.0: ,

3. Use the standard normal table to find the probability between these z-scores.

Try solving on your own before revealing the answer!

Q4. What is the probability that the sample mean will be between 19.5 and 20.5?

Background

Topic: Sampling Distribution of the Mean (Normal Distribution)

This question tests your ability to calculate the probability that the sample mean falls within a narrower range using the sampling distribution.

Key Terms and Formulas:

• Standard error:

• Z-score:

• Probability: Use the standard normal table for

Step-by-Step Guidance

1. Calculate the standard error as before.

2. Find the z-scores for 19.5 and 20.5.

3. Use the standard normal table to find the probability between these z-scores.

Try solving on your own before revealing the answer!

Q5. A company needs to study the average wage of mid-level managers. You assume the standard deviation is $15 per hour. What sample size would you need to be 95% confident that you are within $5 of the true wage?

Background

Topic: Sample Size Calculation for Confidence Intervals

This question tests your ability to determine the required sample size for a desired margin of error and confidence level.

Key Terms and Formulas:

• Margin of error ():

• Sample size formula:

• 95% confidence:

Step-by-Step Guidance

1. Identify the desired margin of error (), standard deviation (), and confidence level (95%).

2. Find the z-score for 95% confidence ().

3. Set up the sample size formula:

Try solving on your own before revealing the answer!