Chapter 8 Notes Section 8-1 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 samples or samples. Examples include studies on identical twins, or before and after studies done on the same members. To test two hypotheses of two large (n 30) samples, you must first write the null and alternate hypotheses. H 0 usually states that there is difference between the parameters of two populations. The H 0 ALWAYS contains an element of equality (,, =) The H a is the complement of the H 0. 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 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. Section 8-2 To test two hypotheses of two small (n < 30) 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 are equal, we combine information from the two samples to calculate a estimate of the σ. If the population variances are 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. 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 are or. If the variances are equal, the estimate is pooled ( ), otherwise it s not ( ). Identify H 0 and H a Identify α P-value Test for a 2 Sample z-test (Large Independent Samples) n > 30 Determine whether samples are. If, can t do it (Yet) If, run STAT Test Find p If p α, Reject H 0 If p α, Fail to reject H 0 Find z or Identify H 0 and H a Identify α If Variances are ( ) If Variances are P-value Test for a 2 Sample t-test (Small Samples) n < 30 STAT Test Find p If p α, Reject H 0 If p α, Fail to reject H 0 Find t ( ), the data is, the data is
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, since they are ( ). 2) Sample 1: test scores for 35 statistics students. Sample 2: test scores for 42 biology students who do not study statistics. Samples are ; they have to do with each other and the sample sizes are. 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. Females Males x 1 = $2290 x 2 = $2370 s 1 = $750 s 2 = $800 n 1 = 200 n 2 = 200 What are the hypotheses, and which one represents the claim? H 0 : H a : (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? -tailed. STAT Test 3: Enter the information asked for, being careful to put values in the correct places. σ 1 = ; σ 2 = ; x 1 = ; n 1 = x 2 = ; n 2 = ; Select for a -tail test At the 5% significance level, there is evidence to support the organization s claim that there is a difference in the mean credit card debt of males and females. 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? Texas Virginia x 1 = $248 x 2 = $252 s 1 = $15 s 2 = $22 n 1 = 50 n 2 = 35 What are the hypotheses, and which one represents the claim? H 0 : H a : (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? -tailed. STAT Test 3: Enter the information asked for, being careful to put values in the correct places. σ 1 = ; σ 2 = ; x 1 = ; n 1 = x 2 = ; n 2 = ; Select for a -tail test
At the 1% significance level, there is 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. Example 1 (Page 454) The braking distances of 8 Volkswagen GTIs and 10 Ford Focuses were tested when traveling at 60 miles per hour 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. GTI Focus x = 134 ft x = 143 ft s 1 = 6.9 ft s 1 = 2.6 ft n 1 = 8 n 1 = 10 Write the hypotheses The claim is that there is a difference between the stopping distances, or that they are NOT equal. H 0 : H a : (claim) This is a -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 Enter the,, and for each sample. Enter,,,,,, in that order. Designate a -tailed test. The variances are NOT equal (given in the problem), so select for Pooled. t = p = NOTE: This test is SO close that if you used the rejection region test, like the book does, you would actually fail to reject the null by a margin of 3 one-thousandths. 2 nd 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. 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. Manufacturer Competition x = 1275 ft x = 1250 ft s 1 = 45 ft s 1 = 30 ft n 1 = 14 n 1 = 16 Write the hypotheses Since the claim is that the manufacturer s range than its competitor s, the hypotheses are: H 0 : (claim) H a : This is a -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 STAT Test 4 Enter the,, and for each sample. Enter,,,,,, in that order. Designate a -tailed test.
The variances are equal (given in the problem), so select for Pooled. t = ; p = Section 8-3 Testing the Difference Between Means (Dependent Samples) In Sections 8-1 and 8-2, we did two-sample hypothesis testing with independent samples using the test statistic x 1 x 2 (the difference between the means of the two samples). To do a two-sample hypothesis test with dependent sample, you will use a different technique. You have to find the for each data pair (remember, in a dependent sample, every member of one sample has a in the other sample). d = x 1 x 2 The test statistic is the of all the (d ) (the total of all the divided by the of ). We can conduct the hypothesis test on two dependent samples if ALL of the following conditions are met: 1) The samples must be randomly selected. 2) The samples must be dependent (paired). 3) Both populations must be normally distributed. If all of these conditions are met, we will use a t-distribution with n 1 degrees of freedom (n is the number of data pairs). Steps to Using the t-test for the Difference Between Means (Dependent Samples) State H 0 and H a. Identify α (Given to you) Find the differences between data pairs STAT Edit Put one set of data into L1 and the other set into L2 Highlight L3 and enter L1 L2 to get the differences into L3 Run STAT Test (from Chapter 7) on L3 (the differences column) Select Data, enter the mean in the hypotheses (usually 0), designate L3 and whether it is left, right, or twotailed. Make a decision to reject or fail to reject the null hypothesis. If p α, reject the null. If p > α, fail to reject the null. Section 8-3 Examples Example 1 (Page 463) A golf club manufacturer claims that golfers can lower their scores by using the manufacturer s newly designed golf clubs. Eight golfers are randomly selected, and each is asked to give his or her most recent score. After using the new clubs for one month, the golfers are again asked to give their most recent score. The scores for each golfer are shown in a table below. Assuming the golf scores are normally distributed, is there enough evidence to support the manufacturer s claim at α = 0.10? Golfer 1 2 3 4 5 6 7 8 Old Clubs 89 84 96 82 74 92 85 91 New Clubs 83 83 92 84 76 91 80 91
Step 1: Write your hypotheses: The claim is that the scores will go down, which means that x 1 x 2 should be positive. This means that the mean of the differences will be than zero. H a:, which makes the null hypothesis H 0:. The hypothesis is the claim. Since H a says, we have a tailed test. Step 2: Identify α: (given) Step 3: Find the data pair differences. STAT Edit L1 and L2, then subtract into L3. Step 4: Use STAT Test 2 on L3, with the mean in the hypotheses being zero. Enter 0 for μ 0, designate L3 and a tail test. You get a standardized test statistic of t =, and p = Step 5: Make a decision () Example 2 (Page 464) A state legislator wants to determine whether her performance rating (0 100) has changed from last year to this year. The following table shows the legislator s performance rating from the same 16 randomly selected voters for last year and this year. At α = 0.01, is there enough evidence to conclude that the legislator s performance rating has changed? Assume the performance ratings are normally distributed. Voter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Last 60 54 78 84 91 25 50 65 68 81 75 45 62 79 58 63 This 56 48 70 60 85 40 40 55 80 75 78 50 50 85 53 60 Step 1: Write your hypotheses: The claim is that the scores will be different. This means that the mean of the differences will be than zero. H a:, which makes the null hypothesis H 0:. The hypothesis is the claim. Since H a says, we have a tailed test. Step 2: Identify α: (given) Step 3: Find the data pair differences. STAT Edit L1 and L2, then subtract into L3. Step 4: Use STAT Test 2 on L3, with the mean in the hypotheses being. Enter 0 for μ 0, designate L3 and a tail test. You get a standardized test statistic of t =, and p = Step 5: Make a decision () We should the null hypothesis and conclude that the legislator s rating has from last year to this year. Section 8-4 Testing the Difference Between Proportions Always assume that there is between the population proportions (p 1 = p 2) To use a z-test to test for differences in proportions, 3 conditions must be met: 1) Samples must be randomly selected.
2) Samples must be independent. 3) np and nq must both be for samples. n 1 p 1 5, n 1 q 1 5, n 2 p 2 5, n 2 q 2 5. On the calculator, you would use STAT-TEST-6 (2-PropZTest). Use p 1 n 1 for x 1 and p 2 n 2 for x 2. (Unless n 1 and n 2 are given to you). Make certain that your x values are ; the calculator will not accept as x values. Round to the. Once you have the p-value, make your decision. Section 8-4 Examples Example 1 (Page 473) In a study of 200 randomly selected adult female and 250 randomly selected adult male Internet users, 30% of the females and 38% of the males said that they plan to shop online at least once during the next month. At α = 0.10, test the claim that there is a difference between the proportion of female and the proportion of male Internet users who plan to shop online. Write our hypotheses and identify the claim. Since the claim is that there is a difference, we are saying that they are. H 0 : H a : (claim) This is a -tailed test. STAT Test 6 x 1 = p 1(n 1 ) =.3 200 = ; n 1 = ; x 2 = p 2(n 2 ) =.38 250 = ; n 2 = Designate a -tail test ( ). Make a decision () We should the null hypothesis and conclude that there a difference between the proportion of female and male Internet users who plan to shop online. (At least at the 10% significance level). Example 2 (Page 474) A medical research team conducted a study to test the effect of a cholesterol-reducing medication. At the end of the study, the researchers found that of the 4700 randomly selected subjects who took the medication, 301 died of heart disease. Of the 4300 randomly selected subjects who took a placebo, 357 died of heart disease. At α = 0.01, can you conclude that the death rate due to heart disease is lower for those who took the medication than for those who took the placebo? Write the hypotheses and identify the claim. The claim is that the proportion of heart disease deaths is with the medication than it is without it. H 0 : H a : (claim) This is a -tailed test. STAT Test 6 x 1 = ; n 1 = ; x 2 = ; n 2 = Designate a -tail test ( ). Make a decision () We should level of significance). the null hypothesis and conclude that the death rate due to heart disease is for those who took the medication than for those who took the placebo (at least at a 1%