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[answered] 1. (10pts) The table below shows the hours of relief provid


I am working on an assignment that involves ANOVA, regression analysis, hypothesis testing, and tests like Wilcox and Sign test. I am trying to do the assignment in R and I am aware of most of the functions that can be applied. I need help in understanding how to answer the questions / what information is required from the questions and how to choose the right method. I need help with questions (3, 5, 6 ) of attached document and I need to submit it by Tuesday night.


1. (10pts) The table below shows the hours of relief provided by two analgesic drugs in

 

12 patients suffering from arthritis. Is there any evidence that one drug provides

 

longer relief than the other?

 

case

 

1

 

DrugA 2

 

DrugB 3.5 2

 

3.6

 

5.7 3

 

2.6

 

2.9 4

 

2.6

 

2.4 5

 

7.3

 

9.9 6

 

3.4

 

3.3 7

 

14.9

 

16.7 8

 

6.6

 

6.0 9

 

2.3

 

3.8 10

 

2.0

 

4.0 11

 

6.8

 

9.1 12

 

8.5

 

20.9 a) (5pts) Check the assumption and choose the appropriate test

 

b) (5pts) Perform the test, manually or using R, you chose in part (a) to test

 

1. (20pts) The body temperatures are provided for a sample of n = 65 healthy men and n =

 

65 health women. Do men and women have the same body temperature on average?

 

Analyze this data set using R and do the following parts to answer this question. Using

 

the following code(copy&paste) to load the data to R:

 

maletemp<c(96.3,96.7,96.9,97,97.1,97.1,97.1,97.2,97.3,97.4,97.4,97.4,97.4,97.5,97.5,97.6,97.6,97.6,97.7,97.8,97.8,

 

97.8,97.8,97.9,97.9,98,98,98,98,98,98,98.1,98.1,98.2,98.2,98.2,98.2,98.3,98.3,98.4,98.4,98.4,98.4,98.5,9

 

8.5,98.6,98.6,98.6,98.6,98.6,98.6,98.7,98.7,98.8,98.8,98.8,98.9,99,99,99,99.1,99.2,99.3,99.4,99.5)

 

femaletemp<c(96.4,96.7,96.8,97.2,97.2,97.4,97.6,97.7,97.7,97.8,97.8,97.8,97.9,97.9,97.9,98,98,98,98,98,98.1,98.2,98

 

.2,98.2,98.2,98.2,98.2,98.3,98.3,98.3,98.4,98.4,98.4,98.4,98.4,98.5,98.6,98.6,98.6,98.6,98.7,98.7,98.7,98

 

.7,98.7,98.7,98.8,98.8,98.8,98.8,98.8,98.8,98.8,98.9,99,99,99.1,99.1,99.2,99.2,99.3,99.4,99.9,100,100.8) a) (5pts) In order to test if body temperatures for men and women differ on average,

 

set up the appropriate null and alternative hypothesis. Define the parameters of

 

interest and state H0 and Ha in terms of these parameters.

 

b) (10pts) Suppose we test the hypothesis in part (a) using a significance level ? =

 

0.05. Assuming the variances for men and women are equal, perform a t-test

 

manually or in R to test the hypothesis. State the conclusion of your test in the

 

context of this problem.

 

c) (5pts) Compute a 99% confidence interval for the difference in mean temperatures

 

between men and women. Write a sentence interpreting this estimate of the

 

difference. Is this interval narrower or wider than the 95% confidence interval? Is

 

zero in this interval? Comment on this.

 

3. (10pts) A study was done on the effects of thermal pollution on clams. Clams were

 

collected at three sites: an intake site to a plant, a discharge site and a site near

 

Interstate 55. The goal of this problem is to determine if the clams differ in terms of

 

heights at the three sites. The ANOVA table is provided below. Based on the table, please

 

do the following parts: Source DF Sum of

 

Squares Mean Square F Value Pr > F Model 2 0.57607022 0.28803511 1.62 Error 71 12.62678383 0.17784203 Total 73 13.20285405 0.2052 a) (2pts) To test if the mean heights of the clams are equal at the three sites, define the

 

appropriate parameters and state H0 and Ha for this problem.

 

b) (3pts) Read the ANOVA output and verify that the F-test statistic is the ratio of the

 

appropriate mean squares from the ANOVA table. What are the numerator and

 

denominator degrees of freedom for the F-test?

 

c) (2pts) What is your conclusion using a significance level ? = 0.05? How about ? =

 

0.10?

 

d) (3pts) Does it make sense to do multiple comparisons looking at differences in pairs

 

of mean heights for this problem? Explain.

 

4. (20pts) A carcinogenicity study was conducted to examine the tumor potential of a drug

 

product scheduled for initial testing in humans. A total of 300 rats (150 males and 150

 

females) were studied for a 6 month period. At the beginning of the study, 100 rats (50

 

males and 50 females) were randomly assigned to the control group, 100 to the low-dose

 

group, and the remaining 100 to the high-dose group. On each day of the 6-month period,

 

the rats in the control group received an injection of an inert solution, whereas those in the

 

treatment groups received an injection of the solution plus drug. The sample data are

 

shown in the accompanying table.

 

Rat group

 

Control

 

Low dose

 

High dose Number of Tumors

 

One or more

 

10

 

14

 

19 none

 

90

 

86

 

81 a. (5pts) Give the percentage of rats with one or more tumors for each of the three

 

treatment groups.

 

b. (10pts) Conduct a test of whether there is significant difference in the proportion of

 

rats having one or more tumors for the three treatment groups with alpha=0.05.

 

c. (5pts) Does there appear to be a drug-related problem regarding tumors for this

 

drug product? That is, as the dose is increased, does there appear to be an increase

 

in the proportion of rats with tumor?

 

5. (17pts) The Federal Trade Commission rates different brands of domestic cigarettes. In

 

their study, they measured the amount of carbon monoxide (co) in mg. and the amount of

 

nicotine (mg.) produced by a burning cigarette of each brand. A simple linear regression model was run co as the dependent variable and nicotine as the independent (or predictor)

 

variable. The output is shown below, use this output to answer the questions below. Source

 

Model

 

Error

 

Total DF

 

1

 

23

 

24 Sum of

 

Squares

 

462.25591

 

76.89449

 

539.15040 Root MSE

 

1.82845

 

Dependent Mean 12.52800

 

Parameter

 

Variable DF

 

Intercept

 

nicotine Mean

 

Square

 

462.25591

 

3.34324 R-Square Estimates

 

Parameter

 

Estimate

 

Error

 

1

 

1.66467

 

1

 

12.39541 F Value Pr > F

 

138.27 <.0001 0.8574 Standard

 

0.99360

 

1.05415 t Value 1.68

 

11.76 Pr > |t| 0.1074

 

<.0001 a. (3pts) How many samples did the Federal Trade Commission evaluate?

 

b. (3pts) What percentage of variation in carbon monoxide is explained by the

 

amount of nicotine?

 

c. (3pts) Write down the least-squares regression equation.

 

d. (5pts) Does the amount of carbon monoxide produced by a burning cigarette

 

depend on how much nicotine is in the cigarette? Perform an appropriate

 

hypothesis test to answer this question using the above output. Be sure to define

 

any parameters you use. Base you conclusion on the p-value of the test.

 

e. (3pts) Give an estimate of how much carbon monoxide increases on average for

 

each additional milligram of nicotine in the cigarette.

 

6. (23pts) The following table gives data on the concentration of chlorophyll-a in a lake

 

along with the concentration of phosphorus in the lake (Source: Smith and Shapiro 1981).

 

Chlorophyll-a is used as an indicator of water quality that measures the density algal cells.

 

Phosphorus stimulates algal growth. We want to use a simple linear regression model to

 

model the relationship between chlorophyll-a and phosphorus in the lake. Please analyze

 

the data and answer the questions.

 

Data:

 

Obs chlor phos log(chlor) log(phos)

 

1 95.0 329.0 4.55388 5.79606

 

2 39.0 211.0 3.66356 5.35186

 

3 27.0 108.0 3.29584 4.68213

 

4 12.9 20.7 2.55723 3.03013

 

5 34.8 60.2 3.54962 4.09767

 

6 14.9 26.3 2.70136 3.26957

 

7 157.0 596.0 5.05625 6.39024

 

8 5.1 39.0 1.62924 3.66356 9

 

10

 

11

 

12

 

13

 

14

 

15

 

16

 

17

 

18

 

19

 

20

 

21

 

22

 

23

 

24

 

25 10.6 42.0 2.36085 3.73767

 

96.0 99.0 4.56435 4.59512

 

7.2 13.1 1.97408 2.57261

 

130.0 267.0 4.86753 5.58725

 

4.7 14.9 1.54756 2.70136

 

138.0 217.0 4.92725 5.37990

 

24.8 49.3 3.21084 3.89792

 

50.0 138.0 3.91202 4.92725

 

12.7 21.1 2.54160 3.04927

 

7.4 25.0 2.00148 3.21888

 

8.6 42.0 2.15176 3.73767

 

94.0 207.0 4.54329 5.33272

 

3.9 10.5 1.36098 2.35138

 

5.0 25.0 1.60944 3.21888

 

129.0 373.0 4.85981 5.92158

 

86.0 220.0 4.45435 5.39363

 

64.0 67.0 4.15888 4.20469 ?Q6.txt? provides you the above data. Use the following code to load the data to R.

 

> setwd("C:\\biostat_intro_2015\\final")

 

## change to your specific path where you saved the data

 

> data<-read.table("Q6.txt", header=T) a) (6pts) Provide and examine the scatterplots of chlorophyll-a vs. phosphorus

 

(column 2 and 3) and log(chlorophayll-a) vs. log(phosphorus) (column 4 and 5).

 

Determine whether a linear model adequately describes the relationship between

 

chlorophyll-a and phosphorus or their log transformed data.

 

b) (6pts) Provide the fitted linear regression equation estimate relating whatever you

 

determined in question a.

 

c) (6pts) Perform the t test to check whether the slope of the regression is significant.

 

Write down the null and alternative hypothesis that is being tested with this t-test

 

statistic. Report the p-value and your conclusion.

 

d) (5pts) What is the predicted amount of the average chlorophyll-a given the

 

concentration of phosphorus is 200?

 


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