Inferential Statistics Homework 

 

Show your computations. You may handwrite this assignment, but please write neatly. Or, you could type the written portions and handwrite the questions requiring drawings, etc.  I know there is a lot here, but it is an excellent review [of Chapter 8 and related topics] for your quiz (hint, hint).

 

1.  Why is the "area under the curve" so important in statistics? (Note: You could answer this question with one sentence.)

 

2.  If I have a z-score (representing the score of an individual person) of 1.00 on some measure, how many standard deviations above the mean is my score?

 

3.  Ilene read in the newspaper that the probability of swallowing a spider while you sleep is .05. She subsequently developed insomnia.

 

a.  What is the probability of swallowing a spider in terms of percentages?

b.  Why isn't her insomnia justified?

c.  Why is her insomnia justified?

d.  What does this question have to do with p< .05  in psychological research?

 

4.  We have sent everyone in an introductory psychology class out to check whether people use seat belts.  Each student has been told to look at 100 cars and to count the number of people wearing seat belts. The number found by any given student is considered that students' score, and we are defining the "population" to be all the cars the students observed. The mean score of the class is 30 with a deviation of 7. 

 

a.  Diagram (draw a curve) for this distribution assuming the scores are normally distributed. Label each standard deviation out to 3 standard deviations; label the deviations with scores relevant to the population.

b.  A student who has done very little work all year reported finding 50 seat belt users (out of the 100 supposedly observed).  Do we have reason to suspect that the student just made up a number rather than actually counting?

Hint - We know the population's mean and standard deviation, and we want to know how an individual's score fits in with a population. Which statistical test do we want to use? The formula is in the Appendix of your text. Use the statistical table beginning on page 381 to look up your computed statistic.  Column C in the table ("Area beyond z") represents the likelihood of obtaining that score. 

 

5.  Assume that you found that, during the past year, you spent $2.00 for lunch each day, give or take $0.25.

a.  Draw a rough sketch of this (assume normal) distribution of daily expenditures labeling the mean and labeling each standard deviation out to 3 standard deviations.

b.  If you spend $2.45 on lunch one day, should you worry you were overcharged? (Note: you do not have to do a statistical test to answer this question- use your drawing from above and what we learned about % of scores typically found within each st. dev. on a normal curve.)

c.  Set up a null hypothesis and an alternative or research hypothesis for this situation. Explain your decision to accept or reject the null.

 

6.  Explain the basic logic underlying all statistical tests.  That is, when we run and report conclusions from an F-test, t-test, z-test, etc. -- what, in essence, are we doing/what question are we answering?  An example, in addition to a more general response, might help you get your ideas across. Your answer should be short, I'm NOT looking for a summary of the details of the last 4 lectures.

 

7.  What is meant by the power of a statistical test, and how can it be increased?

 

8.  For each of the following cases, indicate whether the null hypothesis should be accepted or rejected. Given that decision, indicate also the probability of a Type 1 error (where relevant) and a Type 2 error (where relevant).

a.  p = .03  alpha = .01       beta = .30

b.  p = .13  alpha = .05       beta = .30

c.  p = .03  alpha = .05       beta = .70

 

9.  Go to the website described below and "play" with the demonstration.  I've written out some suggestions below.  For this homework, provide answers to the questions a), b), and c) below:

 

Website Information:     www.sc.edu/~west/javahtml/CLT.html

This demonstration will show you graphs of how many spots result when you throw out dice. You control the number of dice thrown (i.e., sample size) and how many times they are thrown before the graph is drawn (i.e., number of samples).

 

First do the following:

1.  Select 1 dice

2.  Select 100 rolls

Is the distribution resulting from rolling 1 “die” 100 times normal? According to the CLT, why or why not?

 

Now try this:

1.  Select 5 dice (larger sample size, analogous to having more subjects in a study).

2. Select 100 rolls

Is the distribution resulting from rolling 5 “die” 100 times normal? According to the CLT, why or why not?

 

Play with other combinations of "sample size" and approximations of "infinite" numbers of samples.

 

a. Why do we care if the sampling distribution of means is normally distributed?

b. What else does the central limit theorem tell us about a population?

c.  Briefly, why is the central limit theorem so important to statistics?