Stat 401AB Exam 1 September 30, 2002

 

1. Sixteen trees are growing in an orchard. The trees were planted in two rows with eight trees in each row, as illustrated in the rudimentary diagram below.

 

T T T T T T T T

T T T T T T T T

 

The 16 trees had all been attacked by a bug that damages the trees. Two chemical treatments (A and B) that could slow and eventually stop the damage were available. Researchers wanted to compare the effectiveness of these two treatments. The researchers randomly assigned the treatments to the trees as follows:

 

A A B B B A A B

B B A A A B B A

 

At the time the chemicals were applied, the trees to the west (left) had sustained the most damage because they were the first trees infected by the bugs. The bugs gradually spread to the east (right) so that the trees on the east had only sustained minor damage at the time the chemicals were applied. Recognizing this, the researchers used a randomization scheme that would guarantee that the two trees lined up in any column running from north to south would receive different treatments.

 

Three weeks after the initial applications of the chemicals, a score indicating the total damage sustained by each tree was assigned by a team of experts. The scores, arranged to correspond to the positions of the trees, are provided below. Larger scores mean more damage.

 

9 7 4 4 3 5 6 2

8 7 6 5 4 4 3 1

 

Is there evidence that one chemical is better than another at preventing damage? Report a test statistic, a two-sided p-value, and a conclusion. Please use the space below and, if necessary, the back of this page for your work. Write your answers at the bottom of the page.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a) test statistic=____________________ b) two-sided p-value=__________________

c) conclusion:

 

2. A researcher tests soil in 9 fields spread across the state of Iowa. From the soil tests, it is determined that 4 of the 9 fields are infested with soybean cyst nematodes. The other 5 fields are free of soybean cyst nematodes. It is well known that soybean cyst nematodes can cause damage to soybean plants and reduce yields. The researcher has developed a new variety of soybeans that he hopes will be resistant to damage from soybean cyst nematodes. The researcher plants the new variety in each of the 9 fields. The yields of each of the 9 fields along with some summary statistics are provided in the table below.

 

Field Type Yield (bu/acre) Mean SD_

Infested : 23, 38, 40, 42 35.75 8.66

Non-infested : 41, 44, 45, 47, 59 47.20 6.94

 

a) In this case, which is the more appropriate tool for determining if the infested fields have significantly lower yield than the non-infested fields the two-sample t-test or the rank-sum test? Why?

 

 

 

 

 

 

b) Conduct the test that you chose as most appropriate in part (a). Use the space below for your work. Write the test statistic, p-value, and your conclusion at the bottom of the page.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

test statistic=____________________ p-value=__________________

conclusion:

3. Problems 1 and 2 each describe a scientific investigation. For each of problems 1 and 2, state whether the investigation was an experiment or an observational study. Explain your reasoning. Also, for any observational study, explain what would need to be done differently to turn the observational study into an experiment.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4. A researcher places a tub filled with pure water on a rooftop for several days. The researcher is interested in measuring the amount of a certain chemical that passes from the air into the tub of water. The tub contains several gallons of water, but the machine that measures the concentration of the chemical works with small samples of at most 10 ml. The researcher randomly selects 5 such samples from the tub and measures the chemical concentration of each sample. The concentrations of the five samples are provided below along with some summary statistics.

 

Sample 1 2 3 4 5 Mean SD

Chemical Concentration 38 34 37 40 36 37 2.24

(in millionths of a gram/liter)

 

a) Provide a 95% confidence interval for the mean concentration of the chemical in the tub.

 

 

 

 

 

 

 

 

 

 

 

b) Mark each statement below as true (T) or false (F).

 

____Approximately 95% of all 10 ml samples from the tub will have a concentration within the limits computed above.

 

____95% is the approximate chance of selecting five 10 ml samples from the tub that will lead to a confidence interval that contains within it limits the mean concentration of the chemical in the tub.

 

____The method used to find the confidence interval above should produce an interval that contains the mean concentration of the chemical in the tub about 95% of time.

5. Suppose the researcher in question 4 actually had data from 6 tubs. Three tubs were randomly placed inside the city limits of a particular city. The other 3 tubs were placed at random locations in a nearly unpopulated section of countryside approximately the same area as the city. As in question 3, five measurements were taken on each tub. The following SAS code and output was used to analyze the data. The researcher concluded that the chemical concentration was significantly higher in the city than in the section of countryside studied. Is this conclusion justified? Provide reasons for your answer.

 

/* location=1 is city. location=2 is country. */

data one;

input location tub concentration;

cards;

1 1 38

1 1 34

1 1 37

1 1 40

1 1 36

1 2 50

1 2 48

1 2 52

1 2 54

1 2 53

1 3 61

1 3 60

1 3 59

1 3 60

1 3 62

2 4 50

2 4 48

2 4 51

2 4 47

2 4 48

2 5 39

2 5 35

2 5 34

2 5 37

2 5 37

2 6 30

2 6 31

2 6 34

2 6 35

2 6 34

;

 

proc ttest;

var concentration;

class location;

run;

 

The TTEST Procedure

Lower CL Upper CL

Variable location N Mean Mean Mean Std Dev Std Err

concentra 1 15 43.981 49.6 55.219 10.147 2.6201

concentra 2 15 35.282 39.333 43.385 7.316 1.889

concentra Diff (1-2) 3.6503 10.267 16.883 8.8458 3.23

 

T-Tests

 

Variable Method Variances DF t Value Pr > |t|

concentration Pooled Equal 28 3.18 0.0036

concentration Satterthwaite Unequal 25.5 3.18 0.0039

 

Equality of Variances

 

Variable Method Num DF Den DF F Value Pr > F

concentration Folded F 14 14 1.92 0.2332