Inferences About Two Means: Independent Samples Section 9-3 Inferences About Two Means: Independent Samples

Key Concept This section presents methods for using sample data from two independent samples to test hypotheses made about two population means or to construct confidence interval estimates of the difference between two population means.

Key Concept In Part 1 we discuss situations in which the standard deviations of the two populations are unknown and are not assumed to be equal. In Part 2 we discuss two other situations: (1) The two population standard deviations are both known; (2) the two population standard deviations are unknown but are assumed to be equal. Because is typically unknown in real situations, most attention should be given to the methods described in Part 1.

Part 1: Independent Samples with σ1 and σ2 Unknown and Not Assumed Equal

Dependent Samples will be seen in Section 9.4 Definitions Two samples are independent if the sample values selected from one population are not related to or somehow paired or matched with the sample values from the other population. Two samples are dependent if the sample values are paired. (That is, each pair of sample values consists of two measurements from the same subject (such as before/after data), or each pair of sample values consists of matched pairs (such as husband/wife data), where the matching is based on some inherent relationship.) Dependent Samples will be seen in Section 9.4 Text will use the wording ‘matched pairs’. Example at bottom of page 469-470 of Elementary Statistics, 10th Edition

Notation 1 = population mean σ1 = population standard deviation n1 = size of the first sample = sample mean s1 = sample standard deviation Corresponding notations for 2, σ2, s2, , and n2 apply to population 2. ***When we have SAMPLE st. dev. (s) we use Student T Distributions! (t* and t) page 470 of Elementary Statistics, 10th Edition

Hypothesis Test for Two Means: Independent Samples Test Statistic for Two Means: Independent Samples USE YOUR “TMEAN” PROGRAM (where 1 – 2 is often assumed to be 0)

Hypothesis Test - cont Degrees of freedom: In this book we use this simple and conservative estimate: df = the smaller of n1 – 1 and n2 – 1. In other words, use whichever sample is smaller, and take one away. Critical values: invT(area to the left, df), with degrees of freedom (df) = the smaller of n1 – 1 and n2 – 1. *left tailed test, α is in the left tail: invT(, df) *right tailed test, α is in the right tail: invT(1- , df) *two tailed test, α is divided equally between the two tails: invT( 𝛼 2 , df) P-values: Use the tcdf feature on your calculator, with degrees of freedom (df) = the smaller of n1 – 1 and n2 – 1. *For right-tailed tests: To find this probability in your calculator, type: tcdf(t test statistic, 99999999, df)   *For left-tailed tests: To find this probability in your calculator, type: tcdf(–99999999, –t test statistic, df) ***Don’t forget if your test is two-sided, double your P-value.***

Example 1: A headline in USA Today proclaimed that “Men, women are equal talkers.” That headline referred to a study of the numbers of words that samples of men and women spoke in a day. Given below are the results from the study. Use a 0.05 significance level to test the claim that men and women speak the same mean number of words in a day. Does there appear to be a difference? a) State the null hypothesis and the alternative hypothesis. H0: μ1 = μ2 H1: μ1 ≠ μ2 b) What is/are the critical value(s)? Since this test is two-tailed, find 𝛼 2 . The degrees of freedom is 185 (186 is the smaller sample. n – 1 = 185) Type into your calculator: invT(0.025, 185) .05 2 =0.025 z* = ±1.973

Type into your calculator: tcdf(–99999, –0.68, 185) c) Calculate the test statistic. Use the TMEAN program: d) What is the P-value? Type into your calculator: tcdf(–99999, –0.68, 185) Since this test is two-tailed, don’t forget to DOUBLE the p –value! e) What is the conclusion? Compare the p-value to . Fail to reject H0. There is NOT enough evidence to suggest that men and women DO NOT speak the same mean number of words in a day. The calculator will ask for these 6 pieces of data t = –0.676 = 0.250×2 = 0.500 0.5 > 0.05 Make sure your context matches your alternative hypothesis; not the null hypothesis!

Example 2: A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure, measured in mm/Hg, by following a particular diet. Use a significance level of 0.01 to test the claim that the treatment group is from a population with a smaller mean than the control group. a) State the null hypothesis and the alternative hypothesis. H0: μ1 = μ2 H1: μ1 < μ2 b) What is/are the critical value(s)? Since this test is left-tailed,  is in the left tail The degrees of freedom is 100 (101 is the smaller sample. n – 1 = 100) Type into your calculator: invT(0.01, 100) Treatment Group Control Group n1 = 101 n2 = 105 s1 = 17.4 s2 = 30.2 t* = –2.364

c) Calculate the test statistic. Use the TMEAN program: d) What is the P-value? Type into your calculator: tcdf = (–99999, –8.426, 100) e) What is the conclusion? Compare the p-value to . Reject H0. There IS enough evidence to suggest that the treatment group is from a population with a smaller mean than the control group. Treatment Group Control Group n1 = 101 n2 = 105 s1 = 17.4 s2 = 30.2 The calculator will ask for these 6 pieces of data t = –8.426 = 0 0 < 0.01

Example 3: A researcher was interested in comparing the resting pulse rates of people who exercise regularly and of those who do not exercise regularly. Independent simple random samples of 16 people who do not exercise regularly and 12 people who exercise regularly were selected, and the resting pulse rates (in beats per minute) were recorded. The summary statistics are as follows. Use a 0.025 significance level to test the claim that the mean resting pulse rate of people who do not exercise regularly is larger than the mean resting pulse rate of people who exercise regularly. a) State the null hypothesis and the alternative hypothesis. H0: μ1 = μ2 H1: μ1 > μ2 b) What is/are the critical value(s)? Since this test is right-tailed,  is in the right tail The degrees of freedom is 11 (12 is the smaller sample. n – 1 = 11) Type into your calculator: invT(1 – 0.025, 11) Do not exercise regularly Exercise regularly n1 = 16 n2 = 12 s1 = 10.9 beats/min s2 = 8.2 beats/min t* = 2.201

c) Calculate the test statistic. Use the TMEAN program: d) What is the P-value? Type in your caclulator: tcdf = (1.468, 99999, 11) e) What is the conclusion? Compare the p-value to . Fail to reject H0. There IS NOT enough evidence to suggest that the mean resting pulse rate of people who do not exercise regularly is larger than the mean resting pulse rate of people who exercise regularly. The calculator will ask for these 6 pieces of data Do not exercise regularly Exercise regularly n1 = 16 n2 = 12 s1 = 10.9 beats/min s2 = 8.2 beats/min t = 1.468 = 0.085 0.085 > 0.025