Lab 1 Two Population Means

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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)
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