Q1. Is there statistically significant evidence at the 5% significance level that the mean travel time to school for Ontario secondary school students in 2007–2008 is less than 20 minutes?

Background

Topic: Hypothesis Testing for a Population Mean (One-Sample t-Test)

This question tests your ability to conduct a one-sample t-test to determine if the sample provides evidence that the population mean is less than a specified value.

Key Terms and Formulas

• Null Hypothesis ():

• Alternative Hypothesis ():

• Sample mean (): 17 minutes

• Sample standard deviation (): 9.66 minutes

• Sample size (): 35

• Test statistic formula:

Step-by-Step Guidance

1. State the hypotheses:

   • Null hypothesis:

   • Alternative hypothesis:

2. Identify the sample statistics: , , .

3. Calculate the standard error (SE) of the mean:

4. Compute the t-test statistic using:

5. Determine the degrees of freedom () and find the critical value for a one-tailed test at the 5% significance level.

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Q2. What is the probability that the sample mean amount spent per customer (from a sample of 30) is between and the population standard deviation is ?

Background

Topic: Sampling Distribution of the Sample Mean (Central Limit Theorem, Normal Probability)

This question tests your understanding of the sampling distribution of the sample mean and how to calculate probabilities using the normal distribution.

Key Terms and Formulas

• Population mean ():

• Population standard deviation ():

• Sample size (): $30$

• Standard error of the mean (SE):

• Z-score formula for the sample mean:

Step-by-Step Guidance

1. Calculate the standard error (SE) of the sample mean:

2. Find the Z-scores for and using:

3. Calculate the Z-score for .

4. Calculate the Z-score for .

5. Use the standard normal table to find the probability that is between and by finding .

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Q3. Construct and interpret a 95% confidence interval for the true average transaction amount, given a sample mean of $82.50, sample standard deviation of $16.00, and sample size of 64.

Background

Topic: Confidence Intervals for a Population Mean (with Sample Standard Deviation)

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

Key Terms and Formulas

• Sample mean ():

• Sample standard deviation ():

• Sample size (): $64$

• Standard error (SE):

• Critical value for 95% confidence (z*): approximately 1.96 (since n is large)

• Confidence interval formula:

Step-by-Step Guidance

1. Calculate the standard error (SE):

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

3. Multiply the standard error by the critical value to find the margin of error (ME):

4. Set up the confidence interval using:

5. Interpret the confidence interval in the context of the problem (what does it mean about the true average transaction amount?).

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