Answering Four Questions on Two-Sample Inferential Statistics

Please answer each of the following questions in order. There are four questions in total.

You may write your answers neatly by hand or type them up, or a mix of the two. You will need to submit your assignment as a PDF file. You can use your phone to scan written work and convert to PDF.

You may use any software or technology to answer the questions below. You may also work by hand. I recommend (but do not require) that you use Microsoft Excel.1

Your assignment must be submitted on Moodle. Emailed submissions will not be accepted.

Note: Throughout this assignment, where appropriate, assume unequal variances. That is, use the formulas in our textbook, where the variances are not pooled.

Assuming the observations are normally distributed, what is the minimum value that x₂ can take on that will result in a statistically significant result at the a = 0.05 level?

To solve this problem, work with the formula for a t-test. Find the appropriate critical value for t. (Pay attention to whether you need a positive or negative t– draw the distributions and shade the region corresponding to the p −value.)

Substitute the critical value into the t −test formula, along with all known information, and solve for x₂. Show your working and explain steps as necessary.

2. Thinking about experimental design and the ability to choose hypotheses before testing: Why might a matched-pairs design/test be preferable to a two-sample test? Why might it not be preferable?

Answer in a brief paragraph. You can address both statistical and practical (i.e. real-world resource) considerations in your response.

3. To test the hypothesis that “test anxiety” is worse before the test than during it, 36 subjects are assigned to take a standardized test while wearing a heart rate monitor. Their heart rate is recorded in the minute immediately prior to the test starting, and at the halfway point. It is found that the mean difference is x_d = 8.2 with a standard deviation of 17.0. It is assumed that the observations are Normally distributed.

a. State the hypotheses for the test.

b. What is the distribution for the test? State the type of distribution (t, Normal, etc) as well as its parameters (df for t, mean and standard deviation for Normal).

c. What is the value of the test statistic?

d. What is the p −value? Are the results significant at the a= 0.01 level?

4. Van wonders if Prince Rupert really does have more days with precipitation than Vancouver. To test this, a year between 1937 and 2021 was picked at random (these are years for which weather vrecords are available). Van downloaded the weather data2 and created the following summary of days with precipitation in each city:

a. Assuming that we can regard a randomly chosen year as a random sample, is there sufficient evidence at the a = 0.05 level to support Van’s hypothesis? Give a detailed response, using the appropriate solution sheet from the text (see Appendix E) as a pattern.

b. Do you believe the assumption in part (a) (regarding random sample) to be valid? Explain why or why not.