AP Statistics / Mr. Hansen

Name: _________________________________

Test #5 (Merged Version, Feb. 25-26, 1999)

General Instructions: Show all work and all answers in the spaces provided. If you need more room, write OVER and continue on the reverse side. Unless otherwise stated, all numeric answers are to be rounded to 3 places after the decimal point.

Part I: Fill-In (25 points)

Fill in each blank with the word, symbol, or phrase that best fits. Although most of the choices will come from the list below, you may need to make slight adjustments in some cases.

alternative hypothesis
confidence intervals
critical value (upper p critical value) for z and t tests
degrees of freedom
distribution-free procedure
matched pairs procedure
nonparametric procedure
null hypothesis
one-sided alternative / one-sided tests
power
P-value

robust procedure
sign test
significance
significance level
standard error
t distribution
t test for significance
two-sided alternative / two-sided tests
Type I error
Type II error
z test for significance

1-4.

The ______________ for matched pairs is an example of a ______________ (sometimes called "distribution-free") procedure that relies on binomial theory. Because this test makes few assumptions about the underlying distribution, it is very ______________ (i.e., insensitive to violations of assumptions). When using this distribution-free test in its simplest form to resolve a one-sided research question, we would use as our alternative hypothesis either Ha: p > 0.5 or Ha: p < 0.5, depending on whether we were trying to infer an increase or a decrease in our variable of interest. (Here p denotes the probability of an increase in going from the first to the second item of each pair.) Meanwhile, our ______________ would be that p equals 0.5.

5.

The t procedures for computing margin of error in a confidence interval rely on multiplying a t critical value by the ______________ . This latter quantity involves s, which is not resistant to outliers. Therefore, t procedures are not robust when used with outliers, especially in small samples.

6-7.

If our SRS has 28 data points, then the ______________ for the t test equals ______________.

8-9.

If a particular value of the ______________ is specified, the power of your test against that alternative equals ______________.

10-11.

A t or z score that exceeds, in the proper direction(s), the critical value for an agreed-upon level a , establishes that the sample mean is ______________ at level a , provided all other assumptions are met. In other words, we are saying that chance alone would produce a sample mean this extreme with a probability of ______________ or less.

12.

A 90% confidence interval for a sample mean drawn from an SRS represents a belief that ________________________________________________________________________

________________________________________________________________________

 

 

Part II: Computation/Lookup (25 points)

Fill in each answer blank. For full credit, you must show the formula or technique you are using. In some cases, your justification will consist of only 2 words: "by calculator" or "from table." However, you must provide it.

13.

In a sample of size 51 drawn from an SRS having mean 24.7 and s = 2.54, a 95% confidence interval for the true mean is ______________ to ______________ . The symbol s denotes population standard deviation.
   
   
   
   
   

14.

The t score corresponding to the 95th percentile of a t distribution for a sample of size 14 is ______________ . Provide a brief explanation.
   
   
   
   
   

15.

Determine the skewness, if any, of the following data points: 35, 38, 40, 41, 43, 43, 44, 47, 49. Support your answer with a normal quantile plot or a stemplot (your choice).
   
   
   
   
   
   
   
   
   

16.

A z statistic of 1.7 corresponds to a one-sided P-value of between ______________ and ______________ (or if you prefer, give the exact value here: ______________). Does this correspond to statistical significance under the usual cutoff criterion? ______________ If this were a 2-sided test with the same z score, would we have statistical significance or not? ______________ . Justify your answers in the space below.

 

 

Part III: Problem Solving (30 points)

Show everything: setup, definitions of variables (if any), assumptions, computations, conclusions, and discussions of shortcomings (if any) in the experimental design.

17.

Assume that SAT math scores among juniors in D.C. in the absence of coaching vary normally with m =475. Describe a matched pairs design to see if 1000 juniors in D.C., when coached, do significantly better after being coached. (You may assume that s for your improvements in scores is 15, and the mean improvement in scores is 3 points.) Write your conclusion and also describe what it means in terms of real-world (i.e., practical) significance. Use reverse side of paper.

18.

An agricultural field trial compares the yield of two varieties of potatoes for commercial use. The researchers divide in half each of 10 small plots of land in different locations and plant each potato variety on one half of each plot. After harvest, they compare the yields in pounds per plant at each location. The 10 differences (variety "Super" minus variety "Normal") give the following statistics: xbar= 0.34, s=0.83. Discuss the experiment, perform a significance test, and answer these questions:

(a) Is there convincing evidence that the "Super" variety has a higher mean yield than the "Normal" variety?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) If sample size were increased to 25, how would the power of the test change against the alternative that the mean difference between varieties ("Super" minus "Normal") is 0.5 pounds per plant? (You may use s =0.5 to perform your computations for n=25, or if you prefer, you may give a qualitative answer if it is well supported.)

 
   
   
   
   
   
   

 

 

Part IV: In-Depth Understanding (20 points)

Write thoughtful answers using complete sentences. Brevity is preferred to padding, however.

19.

A researcher is interested in determining what variables cause people to be more susceptible to catching colds. He or she obtains a large random sample of schoolchildren and randomly divides them into 100 groups of 20 in order to experiment with 100 different behaviors that may or may not have an effect in avoiding colds. In each group of 20, half of the group is randomly assigned to follow one behavior (for example, washing hands 12 times a day) and the other is told to continue behaving with no change from their normal behavior. There are 100 behaviors in all, one for each of the 100 experimental groups. (The 100 control groups do nothing special.) After 6 months of carefully monitoring all the children to be sure that the instructions are followed, the researcher compiles all the data on number of colds and performs a two-sample t test (which we have not studied yet) for significance in each of the groups of 20, following the usual alpha = 0.05 criterion. Six of the behaviors (i.e., 6 of the 100 groups) show statistically significant reductions in the number of colds when the assigned behavior is followed. The researcher enthusiastically recommends that these 6 behaviors should be followed by schoolchildren, since the behaviors have been statistically shown to be effective in reducing colds. Please comment.

 

 
   
   
   
   
   
   
   
   
   
   
   

20.

Does the width of a confidence interval for a fixed level of confidence increase or decrease as n increases? Explain.
   
   
   
   
   
   
   
   

21.

(Simplified version of #19) Explain, using an example if possible, precisely why it is considered improper procedure to conduct a large number of tests looking for significance and then to publish the one or two tests that happen to come up with a low P-value.