Sections 8-1 and 8-2 Independent and Dependent Samples Independent samples have no relation to each other. An example would be comparing the costs of vacationing in Florida to the cost of vacationing in Virginia. Dependent samples are related to each other. Each member of one sample corresponds to a member of the other sample. Also called paired samples or matched samples. Examples include studies on identical twins, or before and after studies done on the same members.

Sections 8-1 and 8-2 Independent and Dependent Samples To test two hypotheses of two large (n ≥ 30) independent samples, you must first write the null and alternate hypotheses. H0 usually states that there is no difference between the parameters of two populations. The H0 ALWAYS contains an element of equality (≤, ≥, =) The Ha is the complement of the H0. If this still doesn’t come easily to you, refer back to the Chapter 7 notes on hypotheses. The steps are the same as what we did in Chapter 7; we just use a different test to find the standardized test statistic. STAT Test 3 will give us the z-score and the p value for the test. Once you have the p-value, everything else is the same as what we did in Chapter 7.

Sections 8-1 and 8-2 To test two hypotheses of two small (n < 30) independent samples, you must use a t-test instead of a z-test. There are two conditions under which you will be conducting two sample t-tests on small samples. If the population variances are equal, we combine information from the two samples to calculate a pooled estimate of the standard deviation 𝝈 . If the population variances are NOT equal, we use the same formula that we used for the z-test above. We will learn how to tell whether they are equal or not in Chapter 10. For now, this information will be given to you.

Sections 8-1 and 8-2 To use the calculator to conduct a two sample t-test, follow these steps. STAT – TESTS – 4 (2-SampTTest) Select Stats, and enter the information that is given to you for each sample. Indicate whether the test is left tailed, right tailed, or two tailed. The Pooled component is simply whether the variances are equal or not. If the variances are equal, the estimate is pooled (Yes), otherwise it’s not (No).

P-value Test for a 2 Sample z-Test (Large Independent Samples) P-value Test for a 2 Sample t-Test (Small Independent Samples) n < 30 Identify H0 and Ha Identify α Determine whether samples are DEPENDENT or INDEPENDENT. If Dependent, can’t do it (Yet) If Independent, run STAT Test 3 Find p If p ≤ α, Reject H0 If p > α, Fail to reject H0 Find z If Variances are EQUAL, the data is POOLED (Yes) If Variances are NOT EQUAL, the data is NOT POOLED (No) STAT Test 4 Find t

Section 8-1 Example 1 (Page 438) Classify each pair of samples as independent or dependent and justify your answer. 1) Sample 1: resting heart rates of 35 individuals before drinking coffee Sample 2: resting heart rates of the same individuals after drinking two cups of coffee. Samples are dependent, since they are related (same individuals). 2) Sample 1: test scores for 35 statistics students. Sample 2: test scores for 42 biology students who do not study statistics. Samples are independent; they have nothing to do with each other and the sample sizes are different.

Section 8-1 Example 2 (Page 442) A consumer education organization claims that there is a difference in the mean credit card debt of males and females in the United States. The results of a random survey of 200 individuals from each group are shown below. The two samples are independent. Do the results support the organization’s claim? Use α = 0.05. What are the hypotheses, and which one represents the claim? 𝐻 0 : 𝜇 1 = 𝜇 2 𝐻 𝑎 : 𝜇 1 ≠ 𝜇 2 (Claim) Another way of saying this is to say that 𝐻 0 : 𝜇 1 − 𝜇 2 =0 and 𝐻 𝑎 : 𝜇 1 − 𝜇 2 ≠0 (Claim) What kind of test are you running? Two-tailed.

Females Males 𝑥 1 =$2290 𝑥 2 =$2370 𝑠 1 =$750 𝑠 2 =$800 𝑛 1 =200 Section 8-1 Example 2 (Page 442) STAT Test 3: Enter the information asked for, being careful to put values in the correct places. 𝜎 1 =750; 𝜎 2 =800; 𝑥 1 =2290; 𝑛 1 =200 𝑥 2 =2370; 𝑛 2 =200; Select ≠ for a two-tail test Females Males 𝑥 1 =$2290 𝑥 2 =$2370 𝑠 1 =$750 𝑠 2 =$800 𝑛 1 =200 𝑛 2 =200

Females Males 𝑥 1 =$2290 𝑥 2 =$2370 𝑠 1 =$750 𝑠 2 =$800 𝑛 1 =200 Section 8-1 Example 2 (Page 442) z = -1.03 p = .3022 Compare p to α. Since .3022 > .05, fail to reject. Females Males 𝑥 1 =$2290 𝑥 2 =$2370 𝑠 1 =$750 𝑠 2 =$800 𝑛 1 =200 𝑛 2 =200

Section 8-1 Example 2 (Page 442) Since we failed to reject the null hypothesis, we failed to support the claim. At the 5% significance level, there is not enough evidence to support the organization’s claim that there is a difference in the mean credit card debt of males and females.

Section 8-1 Example 3 (Page 443) The American Automobile Association claims that the average daily cost for meals and lodging for vacationing in Texas is less than the same average costs for vacationing in Virginia. The table shows the results of a random survey of vacationers in each state. The two samples are independent. At α = 0.01, is there enough evidence to support the claim? What are the hypotheses, and which one represents the claim? 𝐻 0 : 𝜇 1 ≥ 𝜇 2 𝐻 𝑎 : 𝜇 1 < 𝜇 2 (Claim) Another way of saying this is to say that 𝐻 0 : 𝜇 1 − 𝜇 2 ≥0 and 𝐻 𝑎 : 𝜇 1 − 𝜇 2 <0 What kind of test are you running? Left-tailed.

Section 8-1 Example 3 (Page 443) STAT Test 3: Enter the information asked for, being careful to put values in the correct places. 𝜎 1 =15; 𝜎 2 =22; 𝑥 1 =248; 𝑛 1 =50 𝑥 2 =252; 𝑛 2 =35; Select < for a Left-tail test Texas Virginia 𝑥 1 =$248 𝑥 2 =$252 𝑠 1 =$15 𝑠 2 =$22 𝑛 1 =50 𝑛 2 =35

Section 8-1 Example 3 (Page 443) STAT Test 3: Enter the information asked for, being careful to put values in the correct places. z = -.9343 p = .1751 Compare p to α. Since .1751 > .01, fail to reject. Texas Virginia 𝑥 1 =$248 𝑥 2 =$252 𝑠 1 =$15 𝑠 2 =$22 𝑛 1 =50 𝑛 2 =35

Section 8-1 Example 3 (Page 443) Since we failed to reject the null hypothesis, we failed to support the claim. At the 1% significance level, there is not enough evidence to support the American Automobile Association’s claim that the average daily cost for meals and lodging for vacationing in Texas is less than the same average costs for vacationing in Virginia.

Section 8-2 Example 1 (Page 454) The braking distances of 8 Volkswagen GTIs and 10 Ford Focuses were tested when traveling at 60 mph on dry pavement. The results are shown below. Can you conclude that there is a difference in the mean braking distances of the two types of cars? Use α = 0.01. Assume the populations are normally distributed and the population variances are not equal. What are the hypotheses, and which one represents the claim? The claim is that there is a difference between the stopping distances, or that they are NOT equal. 𝐻 0 : 𝜇 1 = 𝜇 2 𝐻 𝑎 : 𝜇 1 ≠ 𝜇 2 (Claim) Two-Tailed Test Another way of saying this is to say that 𝐻 0 : 𝜇 1 − 𝜇 2 =0 and 𝐻 𝑎 : 𝜇 1 − 𝜇 2 ≠0 (Claim)

STAT Test 4 (2-SampTTest) Example 1 (Page 454) 𝐻 0 : 𝜇 1 = 𝜇 2 𝐻 𝑎 : 𝜇 1 ≠ 𝜇 2 (Claim) STAT Test 4 (2-SampTTest) Enter the mean, standard deviation, and sample size for each sample. Enter 134, 6.9, 8, 143, 2.6, 10, in that order. Designate a two-tailed test. The variances are NOT equal (given in the problem), so select No for Pooled. GTI Focus 𝑥 1 =134 𝑓𝑡. 𝑥 2 =143 𝑓𝑡. 𝑠 1 =6.9 𝑓𝑡. 𝑠 2 =2.6 𝑓𝑡. 𝑛 1 =8 𝑛 2 =10

Since we rejected the null, we support the claim. Example 1 (Page 454) t = -3.496 p = .0073 Compare p to α. Since .007 ≤ .01, reject the null. Since we rejected the null, we support the claim. NOTE: This test is SO close that if you used the rejection region, like the book does, you would actually fail to reject the null by a margin of 3 one-thousandths. 2nd VARS 4 (.01/2) with 7 d.f. gives a rejection region to the left of -3.499 or to the right of 3.499. Our t-score of -3.496 is JUST inside of those numbers, so we would fail to reject the null. GTI Focus 𝑥 1 =134 𝑓𝑡. 𝑥 2 =143 𝑓𝑡. 𝑠 1 =6.9 𝑓𝑡. 𝑠 2 =2.6 𝑓𝑡. 𝑛 1 =8 𝑛 2 =10

Manufacturer Competitor 𝑥 1 =1275 𝑓𝑡. 𝑥 2 =1250 𝑓𝑡. 𝑠 1 =45 𝑓𝑡. Example 2 (Page 455) A manufacturer claims that the calling range (in feet) of its 2.4 GHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 randomly selected phones from the manufacturer and 16 randomly selected phones from its competitor. The results are shown below. At α = 0.05, can you support the manufacturer’s claim? Assume the populations are normally distributed and the population variances are equal. Write the hypotheses Since the claim is that the manufacturer’s range is greater than its competitor’s, the hypotheses are: 𝐻 0 : 𝜇 1 ≤ 𝜇 2 𝐻 𝑎 : 𝜇 1 > 𝜇 2 (Claim) This is a right-tailed test. Manufacturer Competitor 𝑥 1 =1275 𝑓𝑡. 𝑥 2 =1250 𝑓𝑡. 𝑠 1 =45 𝑓𝑡. 𝑠 2 =30 𝑓𝑡. 𝑛 1 =14 𝑛 2 =16

Manufacturer Competitor 𝑥 1 =1275 𝑓𝑡. 𝑥 2 =1250 𝑓𝑡. 𝑠 1 =45 𝑓𝑡. Example 2 (Page 455) A manufacturer claims that the calling range (in feet) of its 2.4 GHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 randomly selected phones from the manufacturer and 16 randomly selected phones from its competitor. The results are shown below. At α = 0.05, can you support the manufacturer’s claim? Assume the populations are normally distributed and the population variances are equal. Another way of saying this is to say that 𝐻 0 : 𝜇 1 − 𝜇 2 ≤0 and 𝐻 𝑎 : 𝜇 1 − 𝜇 2 >0 Manufacturer Competitor 𝑥 1 =1275 𝑓𝑡. 𝑥 2 =1250 𝑓𝑡. 𝑠 1 =45 𝑓𝑡. 𝑠 2 =30 𝑓𝑡. 𝑛 1 =14 𝑛 2 =16

Enter the mean, standard deviation, and sample size for each sample. Example 2 (Page 455) Stat Test 4 Enter the mean, standard deviation, and sample size for each sample. Enter 1275, 45, 14, 1250, 30, 16, in that order. Designate a right-tailed test. The variances are equal (given in the problem), so select Yes for Pooled. Manufacturer Competitor 𝑥 1 =1275 𝑓𝑡. 𝑥 2 =1250 𝑓𝑡. 𝑠 1 =45 𝑓𝑡. 𝑠 2 =30 𝑓𝑡. 𝑛 1 =14 𝑛 2 =16

Since we rejected the null hypothesis, we can support the claim. Example 2 (Page 455) t = 1.811 p = .0404 Compare p to α. Since .0404 ≤ .05, reject the null. Since we rejected the null hypothesis, we can support the claim. Manufacturer Competitor 𝑥 1 =1275 𝑓𝑡. 𝑥 2 =1250 𝑓𝑡. 𝑠 1 =45 𝑓𝑡. 𝑠 2 =30 𝑓𝑡. 𝑛 1 =14 𝑛 2 =16

Assignments: Classwork: Pages 444-445, #5-18 All For numbers 13-18, use p-value instead of rejection regions. Pages 456-457, #11-14 All For #11-14, do part b, find p and then do part d. Homework: Pages 445-449, #20-34 Evens Skip part b. Pages 457-459, #16-24 Evens