Practice Midterm 2

This is pulled off of old exams for classes that used a different textbook. So this might not be representative of what you will see for your midterm. In particular, you can expect questions about the 5-step process.

1. The bite of a type of fly that lives in Brazil causes lesions on the skins of those bitten. To measure the size of the lesions, physicians give the bite-victims something called a Montenegro (MTN) Skin Test; a small injection of a substance that causes a splotch to appear on the skin. The area of this splotch is proportional to the area of the lesions. Therefore, by measuring the area of the Montenegro skin test splotch, they can measure the area of the lesions.

1. (5) How would you classify this variable (use all that apply) and why? (Categorical, quantitative, qualitative, continuous, discrete, ordinal.)

b) (5) In order to estimate the mean area of the MTN skin test, a random sample of 35 fly-bite victims was collected. Here is a summary of the MTN areas (in square millimeters). Find a 90% confidence interval for the mean MTN area.

2. Do piano lessons improve the spatial-temporal reasoning of preschool children? Neurobiological arguments suggest that this may be true. A study designed to test this hypothesis measured the spatial-temporal reasoning of 34 preschool children before and after six months of piano lessons. The changes in the reasoning scores are summarized below. (Change is measured as the After score minus the Before score.)

Variable N Average Std. Dev Minimum Median Maximum

reasoning 34 3.6176 3.0552 -3. 4. 9.

Is there evidence that piano lessons improve the spatial-temporal reasoning of preschool children? To answer this question, do the following:

a) Find a 95% confidence interval for the mean change in reasoning scores.

b) State the assumptions you made to get this confidence interval. Do you think the assumptions are good? Based on what evidence?

c) Is your confidence interval evidence that spatial-temporal reasoning was improved by piano lessons? Explain.

3. (5) In the last problem, suppose that you had found that there was a significant change in reasoning scores. Can you conclude that the change was due to piano lessons? If yes, explain why. If no, explain how you would design a study that would allow you to conclude that piano lessons caused the change.

4. An airline knows, through experience, than 10% of the passengers with tickets will not show up for their flight. For this reason, it overbooks flights. For a plane with 100 seats, it will sell 105 tickets. Assume that all ticketed passengers are flying alone and make their decisions about showing up without consulting each other.

a) (5) Let X represent the number of people out of 105 who show up for their flight . What is the distribution of X and why?

b) (5) What's the probability all 105 passengers will show up? (Don't give a number -- just a formula.)

c) (5) Use the normal approximation to find the probability that there will be angry passengers who do not get a seat on the plane.

5. A well known scientific theory states that the rate of coughs at a symphony concert during flu season is lambda=2.5 per minute. Let X represent the number of coughs in a one minute period during a flu-season symphony concert. Suppose that X follows a Poisson distribution.

a) What is the probability of exactly one cough occurring during this period?

b) What is the probability of at least one cough during a one minute period?

c) Based on "theoretical" considerations, do you think the Poisson distribution is a good model for X? Explain.

d) Here are some data from a study that randomly sampled one-minute intervals in a random selection of flu-season concerts. Use this data to estimate lambda.

1

2

1

2

0

1

2

1

5

2

3

5

3

2

0

e) Assume that the Poisson model is a good model for this data. But the exact value of the parameter lambda is in doubt. Do the data above support the hypothesis that lambda=2.5? Write a five-step simulation to provide evidence for or against. Explain how you would make a decision.

6. A bottle-filling machine at the Fizzy Soda bottling and distribution plant is designed to put in 12 ounces of soda into each bottle. In fact, the label on the bottle "guarantees" that it contains 12 ounces of beverage. Because of slight variations, however, the amount it puts in is actually a normal random variable with mean 12.2 ounces and SD 0.2 ounce.

a) What is the probability that a given bottle will have less than 12 ounces?

b) Suppose we take a random sample of 6 bottles. What is the probability that the average amount is less than 12 ounces? Show all steps.