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Final Exam Solutions

1. (8) Write the best answer below the question for the following multiple choice questions. No explanation necessary.

(1) Ten tidal pools of the same size were found in a certain coastal region. It was randomly determined which 5 would receive a treatment (removal of limpets, a type of seaweed grazer) and which 5 would serve as controls. The amount of seaweed covering the floor of the tide pools was measured at the end of the study period. It was desired to see whether the amount of seaweed was affected by limpet removal. Which of the following statements is correct?

a) The data should be analyzed with a two-independent sample t-test. A causal inference can be drawn from the study.

b) The data should be analyzed with a two-independent sample t-test. A causal inference cannot be drawn from the study.

c) The data should be analyzed with a paired t-test. A causal inference can be drawn from the study.

d) The data should be analyzed with a paired t-test. A causal inference cannot be drawn from the study.

A

(2) Which of the following is not an assumption in the ideal model for comparing several populations used for the one-way ANOVA F test?

a) The sample sizes must be equal

b) The populations must all be normally distributed

c) The population variances must be equal

d) The samples for each treatment must be selected randomly and independently

e) All of the above are assumed.

A

(3) Suppose people are randomly assigned into three groups and given either one, two or three mg of a drug. The amount of pain they feel after having their wisdom teeth removed is the Y variable. The amount of drugs is the X variable. A simple regression of Y vs. X is done.

a) If the t-test for X is significant (p-value < .05), then we have evidence that X causes Y to change.

b) No matter what the t-ratio for X is, we cannot determine causation since a regression is being done. To provide evidence for causation, a one-way analysis of variance F test should be done instead.

c) There may be a confounding variable that causes both X and Y to increase. So statistical significance in this problem doesn’t provide evidence for causation.

d) If the t-test for X is insignificant, then we have evidence that X doesn’t cause Y.

e) More than one of the above is true.

A

(4) In a statistical report, the statement is made that the 95% confidence interval for the percentage of babies who are boys is between 51% and 55% (i.e., 53%[pic]2%). This means, that if, in the future, a 95% confidence interval is computed in the same way for each of a large number of random samples of the same size

a) 95% of such intervals will cover (contain) the midpoint 53%

b) 95% of such intervals will cover (contain) the population percentage of boys.

c) 95% of such intervals will overlap (intersect) the interval 51% to 55%

d) 95% of such intervals will completely cover (contain) the interval 51% to 55%

B

(5) A study of human development showed two types of movies to groups of children. Crackers were available in a bowl, and the investigators compared the number of crackers eaten by children watching the different kinds of movies. One kind of movie was shown at 8 A.M. (right after the children had breakfast) and another at 11 A.M. (right before the children had lunch). It was found that more crackers were eaten during the movie shown at 11 A.M. than during the movie shown at 8 A.M. The investigators concluded that the different types of movies had an effect on appetite. The results cannot be trusted because

a) the study was not double blind. Neither the investigators nor the children should have been aware of which movie was being shown.

b) the investigators were biased. They knew beforehand what they hoped to show.

c) the investigators should have used several bowls, with crackers randomly placed in each.

d) the time the movie was shown is a confounding variable.

D

(6) A group of college students believes that herbal tea has remarkable restorative powers. To test its theory, the group makes weekly visits to a local nursing home, visiting with residents, talking with them and serving them herbal tea. After several months, many of the residents are more cheerful and healthy. Which of the following may be correctly concluded from this study?

a) herbal tea does improve one’s emotional state, at least for the residents of nursing homes.

b) there is some evidence that herbal tea may improve one’s emotional state. The results would be completely convincing if a scientist had conducted the study rather than a group of college students.

c) the results of the study are not convincing since only a local nursing home was used and only for a few months.

d) the results of the study are not convincing since the effect of herbal tea is confounded with several other factors.

D

(7) Does taking gingko tables twice a day provide significant improvement in mental performance? To investigate this issue, a researcher conducted a study with 150 adult subjects who took gingko tablets twice a day for a period of six months. At the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Nine of these variables were significantly better (in the sense of statistical significance) at the 5% level for the group taking the gingko tablets as compared to the population as a whole, and one variable was significantly better at the 1% level for the group taking the gingko tablets as compared to the population as a whole. It would be correct to conclude

a) there is good statistical evidence that taking gingko tablets twice a day provides some improvement in mental performance.

b) there is good statistical evidence that taking gingko tablets twice a day provides improvement for the variable that was significant at the 1% level. We should be somewhat cautious about making claims for the variables that were significant at the 5% level.

c) these results would have provided good statistical evidence that taking gingko tablets twice a day provides some improvement in mental performance if the number of subjects had been larger. It is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured.

d) none of the above.

D

(8) Are proficiency test scores affected by the education of the child’s parents? To answer this question, a random sample of 9-year old children was drawn. Each child’s test score and the education level of the parent with the higher level were recorded. The education categories are less than high school, high school graduate, some college, and college graduate. The null hypothesis for the one-way analysis of variance F test is that the population mean test scores are the same for all four education categories. The alternative hypothesis is

a) that the population mean test score is larger for children of college graduates than for the other three educational categories

b) that the population mean test score is smaller for children whose parents both did not graduate from high school than for the other three educational categories

c) that the population mean test score for children of college graduates is larger than the population mean test score for children whose parents both did not graduate from high school

d) none of the above.

D

2. (5) The following is a list of some of the statistical methods you have learned in this course:

A. Two independent samples t-test

B. Matched pairs t-test

C. Methods for comparing several means (One-way analysis of variance F test and Tukey-Kramer adjusted procedure for comparing two of several means)

E. Chi-squared test

F. Simple linear regression

G. Multiple regression

For each of the situations described below, state the technique (from the list above) that you believe is most applicable.

(a) A researcher for OSHA (Occupational Safety and Health Adminstration) wants to see whether cutbacks in enforcement of safety regulations coincided with an increase in work related accidents. For 20 industrial plants, she has the number of accidents in 1980 and 1995.

B

(b) A researcher wants to investigate how fertilizer affects soybean yield. She divides a farm into 30 one-acre plots. Each plot receives a different amount of fertilizer. Soybeans were then planted and the amount of soybeans harvested at the end of the season from each plot were recorded.

F

(c) Can music make you smarter? And if so, which kind of music works best? Two University of California at Irvine professors addressed these questions (as reported on “Dateline” in September 1994). A random sample of 135 students was given tests that measured the ability to reason. One third of the students were then put in a room where rock-and-roll music was played. A second group of 45 students was placed in a room and listened to music composed by Mozart. The last group was placed in a room where no music was played. The students then took a test. The differences (second test score minus first test score) were recorded.

C

(d) A bank would like to develop a model to predict the total sum of money customers withdraw from automatic teller machines (ATMs) on a weekend so that they can be sure to stock an adequate amount of money in each of the machines. They have data on the amount of money withdrawn last weekend for a random sample of 35 ATM machines throughout the city. They believe several factors can be useful in predicting the amount of money withdrawn including the average assessed value of houses in the vicinity of the ATM machine, how far away the nearest branch office of the bank is from the ATM machine, and whether or not the ATM machine is located in a shopping center.

G

(e) There is a theory that the anticipation of a birthday can prolong a person’s life. In a study set up to examine this notion statistically, it was found that only 60 of 747 people whose obituaries were published in Salt Lake City in 1975 died in the three-month period preceding their birthday.

E

3. (6) A randomized experiment is done to measure the effect of a drug on developing mouse’s weights. 10 30-day old mice were randomly divided into groups of five. The drug group received the drug for ten days; the placebo group received a placebo for ten days. The weight gains at the end of the ten days were recorded. JMP output is shown below for two analyses, one is an analysis of how the weight gains for the two groups compare and the other is an analysis of how the log weight gains for the two groups compare.

(a) (3) Assume the additive treatment effect model holds. Find an (approximate) 95% confidence interval for the amount by which taking the drug increases a mouse’s weight gain compared to what the mouse’s weight gain would have been taking the placebo.

From the JMP output for Analysis I, a 95% confidence interval is (-1.274,30.311).

(b) (3) Assume the multiplicative treatment effect model holds. Find an (approximate) 95% confidence interval for the amount by which taking the drug multiplies a mouse’s weight gain compared to what the mouse’s weight gain would have been taking the placebo.

From the JMP output for Analysis II, a 95% confidence interval for the amount by which taking the drug increases a mouse’s log weight gain compared to what the gain would have been from taking the placebo is (0.8019,3.8481). Thus, a 95% confidence interval for the amount by which taking the drug multiplies a mouse’s weight gain compared to what it would have been from taking the placebo is [pic]

(c) (3) Is there strong evidence that taking the drug as opposed to the placebo causes a change in mice’s mean weight gains? Justify your answer using an appropriate test.

The multiplicative model appears more appropriate than the additive model. The box plots for Analysis I show that the drug group has much greater spread than the placebo group; the spreads of the two groups should be about equal if the additive model holds. On the other hand, the box plots for Analysis II show that the drug group and placebo group have about equal spreads on the log scale; this is what we would expect to be the case if the multiplicative model holds. Under the multiplicative model, the t-test of the null hypothesis that the mean of log weight gain for the drug group equals the mean of log weight gain for the placebo group versus the two sided alternative that the means are not the same provides a test of whether taking the drug as opposed to the placebo causes a change in mice’s mean weight gains. From Analysis II, the p-value for this test is .0078. Thus, there is strong evidence that taking the drug as opposed to the placebo causes a change in mice’s mean weight gains.

Analysis I: Y= Weight Gains

Oneway Analysis of Response By Group

[pic]

Means and Std Deviations

|Level |Number |Mean |Std Dev |Std Err Mean |Lower 95% |Upper 95% |

|Drug |5 |16.1461 |15.2061 |6.8004 |-2.735 |35.027 |

|Placebo |5 |1.6276 |1.8113 |0.8100 |-0.621 |3.877 |

t-Test

| |Difference |t-Test |DF |Prob > |t| |

|Estimate |14.519 |2.120 |8 |0.0668 |

|Std Error |6.848 | | | |

|Lower 95% |-1.274 | | | |

|Upper 95% |30.311 | | | |

Assuming equal variances

Analysis II: Y=Log (Weight Gains)

Oneway Analysis of Log Response By Group

[pic]

Means and Std Deviations

|Level |Number |Mean |Std Dev |Std Err Mean |Lower 95% |Upper 95% |

|Drug |5 |2.35527 |1.05441 |0.47155 |1.046 |3.6645 |

|Placebo |5 |0.03029 |1.03417 |0.46250 |-1.254 |1.3144 |

t-Test

| |Difference |t-Test |DF |Prob > |t| |

|Estimate |2.32499 |3.520 |8 |0.0078 |

|Std Error |0.66050 | | | |

|Lower 95% |0.80188 | | | |

|Upper 95% |3.84810 | | | |

Assuming equal variances

4. (10) Lotteries have become important sources of revenue for governments. Many people have criticized lotteries, however, referring to them as a tax on the poor and uneducated. In an examination of the issue, a random sample of 100 adults was asked how much they spend on lottery tickets and was interviewed about various socioeconomic variables. The following data was recorded: amount spent on lottery tickets as a percentage of total household income (Lottery), number of years of education (Education), age (Age), number of children (Children) and personal income in thousands of dollars (Income). The output from multiple regression of Lottery on Education, Age, Children and Income is shown below. Assume the ideal multiple linear regression holds for this problem.

Response Lottery

Summary of Fit

|Rsquare |0.433474 |

|Rsquare Adj |0.40962 |

|Root Mean Square Error |???? |

|Mean of Response |5.39 |

|Observations (or Sum Wgts) |100 |

Analysis of Variance

|Source |DF |Sum of Squares |Mean Square |F Ratio |

|Model |4 |615.4421 |153.861 |18.1722 |

|Error |95 |???? |???? |Prob > F |

|C. Total |99 |1419.7900 | ||t| |

|Intercept | |11.906094 |1.785197 |6.67 ||t| |

|Intercept | |7.5102418 |1.440329 |5.21 |0.0002 |

|Speed | |0.0186022 |0.015311 |1.21 |0.2477 |

[pic]

Observations with Largest Cook’s Distances

|Observation Number |Cook’s Distance |Leverage |

|13 |0.280 |0.257 |

|14 |0.083 |0.204 |

(b) (3) One of the most dangerous contaminants deposited over European countries following the Chernobyl accident of April 1987 was radioactive cesium. To study cesium transfer from contaminated soil to plants, researchers collected soil samples and samples of mushroom mycelia from 17 wooded locations in Umbria, Central Italy, from August 1986 to November 1989. The researchers measured concentrations (Bq/kg) of cesium in the soil and in the mushrooms. The researchers’ goal is to predict Y=concentration in mushrooms based on X=concentration in soil. The output from a simple linear regression is shown below.

The simple linear regression model with these data is not appropriate for drawing inferences. Observation 17 has an enormous Cook’s distance of 10.08. This is much greater than the cutoff of 1 for classifying a point as being influential. Observation 17’s leverage is 0.755>2*(2/17). Thus, observation 17 is influential and has high leverage. We cannot draw reliable inferences over the whole range of explanatory variables (concentration in soil) in the data. We should omit observation 17 from the regression and consider the model to only be reliable for concentration in soil in the range of 0-500.

Bivariate Fit of MUSHROOM By SOIL

[pic]

[pic]

Linear Fit

MUSHROOM = 16.725686 + 0.0959027 SOIL

Summary of Fit

|Rsquare |0.406386 |

|RSquare Adj |0.366812 |

|Root Mean Square Error |36.56475 |

|Mean of Response |44.58824 |

|Observations (or Sum Wgts) |17 |

Analysis of Variance

|Source |DF |Sum of Squares |Mean Square |F Ratio |

|Model |1 |13729.399 |13729.4 |10.2690 |

|Error |15 |20054.718 |1337.0 |Prob > F |

|C. Total |16 |33784.118 | |0.0059 |

Parameter Estimates

|Term | |Estimate |Std Error |t Ratio |Prob>|t| |

|Intercept | |16.725686 |12.41954 |1.35 |0.1981 |

|SOIL | |0.0959027 |0.029927 |3.20 |0.0059 |

[pic]

Observations with Largest Cook’s Distances

|Observation Number |Cook’s Distance |Leverage |

|16 |0.081 |0.082 |

|17 |10.08 |0.755 |

6. (7) A study was conducted to test the effects of a nonsteroidal anti-inflammatory drug on pain. The experiment examines the effects of giving the treatment both before and after surgery or after surgery only. 30 patients between the ages of 18 and 55 undergoing elective knee arthroscopy were enrolled in the study two weeks before their surgery and randomly divided into three groups. Group A received the nonsteroidal anti-inflammatory drug (NSAID) both 3 days prior to surgery and 5 days after surgery. Group B received a placebo before surgery and the NSAID after surgery. Group C received the placebo both before and after surgery. Post-operatively all patients were given prescriptions for codeine which could be taken every 4 to 6 hours as needed. Pain scores were recorded at the time of enrollment in the study (two weeks before surgery), one day before surgery and one week after surgery (higher pain scores indicate that the patient is in more pain). Shown below are JMP analyses for three outcomes – (I) pain at time of enrollment in the study (two weeks before surgery); (II) pain one day before surgery; and (III) pain one week after surgery.

(a) (2) Is there strong evidence that that not all of the treatments are equally effective one week after surgery, i.e., that some of the treatments have higher mean pain scores than other treatments one week after surgery? State this question in terms of a hypothesis test and answer the question of interest, using a test at the 0.05 level.

Let [pic] be the mean pain scores for treatments A, B and C 1-week after surgery respectively. To test [pic] vs. [pic]not all treatments have same mean one week after surgery, we use the one-way Analysis of Variance F-test for Analysis III. The p-value for this test is |t| |

|Intercept | |28.229813 |2.174222 |12.98 ||t| |

|Intercept | |58.325987 |7.374532 |7.91 | ................
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