Lab 1 Two Population Means
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411
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Apr 3, 2024
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Math 411 Lab 1: Inferences Concerning Two Population Mean 1
Overview In this lab, we’ll investigate the ?
-test for two independent random samples and the Mann-Whitney U test. Complete all of the exercises for your lab report. As always, when performing a statistical test, be sure to include all relevant information such as the null hypothesis, value, etc. Also, include your R code/scripts and output for each problem in your lab report. 2
𝒕
-test for Two Independent Random Samples As with the single sample tests, R can be easily used to compute two-sample hypotheses tests. These two-sample tests include the ?
-test and the Mann-Whitney U test. Both tests have default corrections in R to handle potential problems with the data. The R code for two sample ?
-tests when your data is in two columns is: t.test(dataset$variable1, dataset$variable2, var.equal=TRUE and for separate columns for group identities and values: t.test(dataset$variable~dataset$group, var.equal=TRUE) Exercises: 1.
Random samples of largemouth bass and smallmouth bass were taken from a lake and their lengths (in millimeters) were determined. We wish to know if the mean standard length differs between the two species in this lake. The results were as follows. Largemouth Bass Smallmouth Bass 𝑥̅
272.8 164.8 ?
96.4 40.0 𝑛
125 97 a.
State the null and alternative hypotheses. b.
Use a two-sample ?
-test to examine the hypotheses. c.
Compute a 95% confidence interval for the difference of the means.
d.
Interpret your results for parts b and c. Are the results consistent? Explain. e.
Generate random samples from each population using the statistics in the table. Using R, run the two-
sample t
-test and explain the output. 2.
Data on reaction time (in milliseconds) for random samples of 58 men and 68 women were collected. We wish to know is there is a difference in reaction time between men and women in this population. The results were as follows.
Men Women 𝑥̅
170.21 181.31 ?
32.643 45.988 𝑛
58 68 a.
State the null and alternative hypotheses. b.
Use a two-sample ?
-test to examine the hypotheses. c.
Compute a 95% confidence interval for the difference of the means.
d.
Interpret your results for parts b and c. Are the results consistent? Explain. e.
Generate random samples from each population using the statistics in the table. Using R, run the two-
sample t
-test and explain the output. 3.
Six randomly selected pea plants were treated with the plant growth regulator auxin and six randomly selected plants were not treated. We wish to know if the growth regulator affects internode growth (measured in mm). Treated Plants Untreated Plants 15.2 13.5 12.3 9.8 11.6 10.2 14.8 8.7 10.0 9.2 14.2 9.0 a.
State the null and alternative hypotheses. b.
Find the means and standard deviations for each sample and use a two-sample ?
-test to examine the hypotheses. c.
Compute a 95% confidence interval for the difference of the means.
d.
Interpret your results for parts b and c. Are the results consistent? Explain. e.
Using R, run the two-sample t
-test on the data in the table and explain the output. 4.
Using the data in Digital Appendix 3, determine if women smokers have faster pulse rates than women non-
smokers. a.
State the null and alternative hypotheses. b.
Find the means and standard deviations for each sample and use a two-sample ?
-test to examine the hypotheses. c.
Compute a 95% confidence interval for the difference of the means.
d.
Interpret your results for parts b and c. Are the results consistent? Explain. e.
Using R, run the two-sample t
-test on the data in the table and explain the output. 3
Mann-Whitney U Test The nonparametric equivalent to the two-sample ?
-test is the Mann-Whitney U test, which is equivalent to the Wilcoxon rank sum test. As with the two sample ?
-tests, the data to be compared may be arranged in two columns: wilcox.test(dataset$variable1, dataset$variable2, correct=FALSE)
or as a column of group IDs and a column of values: wilcox.test(dataset$variable~dataset$group, correct=FALSE) Exercises: 1.
Male hoop snakes, upon encountering one another, may engage in a protracted ritualized combat behavior until one establishes himself as dominant over the other. We would like to know if these encounters last longer in the presence of a female. Twenty-four males were randomly assigned to pairs. Six randomly selected pairs were tested in the presence of a female and six were tested in the absence of a female. This variable is probably not normally distributed. The results were as follows.
Interaction Time (in minutes) Pairs without Female Pairs with Female 10 59 15 35 8 70 30 65 1 43 80 90 a.
State the null and alternative hypotheses. b.
Perform the Mann-Whitney U test by hand and interpret the results. c.
Using R, run the Wilcoxon rank sum test on the data and explain the output. 2.
Seven tomato plants were treated with chlorogenic acid to determine if this would influence the activity of the enzyme o
-diphenol oxidase in the leaves. Enzyme activity was measured as 𝜇
l O
2
/mg protein/minute. A control group of seven plants was not treated. We do not know if this variable is approximately normally distributed, nor is it possible to determine this with a sample of this small a size. Does this treatment affect activity of the enzyme? Treated Untreated 35 10 45 18 36 8 11 29 41 17 29 8 38 11 a.
State the null and alternative hypotheses. b.
Perform the Mann-Whitney U test by hand and interpret the results. c.
Using R, run the Wilcoxon rank sum test on the data and explain the output.
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