SIN502S Week 6

Test for differences in means for three or more independent populations (ANOVA) Analysis of Variance (ANOVA) is an extension of the two sample t-test used to compare the equality of three or more means. It is based on comparing the variance (or variation) between the data samples to variation within each particular sample. If the between variation is much larger than the within variation, the means of different samples will not be equal. If the between and within variations are approximately the same size, then there will be no significant difference between sample means.

Cont… Assumptions of ANOVA: (i) All populations involved follow a normal distribution. (ii) All populations have the same variance (or standard deviation). (iii) The samples are randomly selected and independent of one another.

ANOVA H0: (the means are all equal) H1: At least one of the means is different Reject H0 if where and ; ; Note:

Examples The National Transportation Safety Board (NTSB) wants to examine the safety of compact cars, midsize cars, and full-size cars. It collects a sample of three for each of these car types. Using the hypothetical data provided below, test whether the mean pressure applied to the driver’s head during a crash test is equal for each types of car. Use alpha= 0.05. Compact cars Midsize cars Full-size cars 643 469 484 655 427 456 702 525 402

Post hoc test Determining which mean(s) is different. If you have failed to reject the null hypothesis in an ANOVA then you are done. However, if you have rejected the null hypothesis then you must conduct a separate test to determine which mean(s) is different. Least Significant Difference Test (LSDT)

Cont… Thus if the absolute value of the difference between any two treatment means is greater than the LSD value, we may conclude that they are not statistically equal.

Cont… To investigate the real cost of owning different makes and models of new automobiles, a consumer protection agency followed 8 owners of new vehicles of four popular makes and models, call them TC, HA, NA, and FT, and kept a record of each of the owner’s real cost in dollars for the first five years. The five-year costs of the 8 car owners are given below: Test at the 10% level of significance whether the data provide sufficient evidence to conclude that there are differences among the mean real costs of ownership for these four models. TC HA NA FT 8423 7776 8907 10333 7889 7211 9077 9217

Homework The following three random samples are taken from three normal populations with respective means μ1, μ2, and μ3, and the same variance σ2. At the 10% level of significance, is there evidence that there are differences in means among the three samples? Sample 1 Sample 2 Sample 3 2 3 5 1 7

Cont… The Mozart effect refers to a boost of average performance on tests for elementary school students if the students listen to Mozart’s chamber music for a period of time immediately before the test. An elementary school teacher conducted an experiment by dividing her third-grade class of 15 students into three groups of 5. The first group was given an end-of-grade test without music; the second group listened to Mozart’s chamber music for 10 minutes; and the third groups listened to Mozart’s chamber music for 20 minutes before the test. The scores of the 15 students are given below: At α=0.10, is there sufficient evidence in the data to suggest that the Mozart effect exists? Group 1 Group 2 Group 3 80 79 73 63 82 74 71 77 70 81 84

Tests concerning variances: one variance and two variances Testing for a single population variance Assuming that the population is normally distributed, we use the Chi-square test to test whether the population variance or standard deviation is equal to a specified value. 1. 2.

Cont… 3.

Examples A manufacturer of candy must monitor the temperature at which the candies are baked. Too much variation will cause inconsistency in the taste of the candy. Past records show that the standard deviation of the temperature has been A random sample of 30 batches of candy is selected, and the sample standard deviation of the temperature is At the 0.05 level of significance, is there evidence that the population standard deviation has increased above ?

Cont… Random sample from a normal population produced the following results. Sample: 18, 15, 19, 31, 49, 12, 48, 29, 45, 50 Is there enough evidence at a 10% significance level to conclude that the population standard deviation is 15.31?

Homework The marketing manager of a branch office of a local telephone operating company wants to study characteristics of residential customers served by her office. In particular, she wants to estimate the mean monthly cost of calls within the local calling region. In order to determine the sample size necessary, she needs an estimate of the standard deviation. On the basis of her past experience and judgment, she estimates that the standard deviation is equal to $12. Suppose that a small-scale study of 15 residential customers indicates a sample standard deviation of $9.25. (a) At the 0.10 level of significance, is there evidence that the population standard deviation is different from $12? (b)What assumption do you need to make in order to perform this test?

Testing for difference between two population variances Two equal variances would satisfy the equation which is equivalent to . Since sample variances are related to chi-square distributions, and the ratio of chi-square distributions is an F-distribution, we can use the F-distribution to test against a null hypothesis of equal variances. Case 1 Case 2 Case 3 We force the test to be right-tailed by keeping the sample with the large variance as our numerator.

Cont… . Note: The sample variance that is larger must always be our numerator and the one with smaller variance becomes the denominator.

Examples A carpet manufacturer company is interested in the time it take customers to receive carpets that were ordered from the plant. Data concerning a sample of delivery times from two branch stores are summarized as follows: At the 0.1 level of significance, is there evidence of a difference in the variances of the shipping time between the two outlets? Sample A Sample B Mean 34.3 days 43.7 days Standard deviation 2.4 days 3.1 days Sample size 41 31

Cont… Random samples from two normally distributed populations produced the following results: Sample A: 27, 52, 41, 20, 33, 59, 41, 28, 29 Sample B: 18, 15, 19, 31, 12, 29, 45 Is there evidence at the 5% significance level to conclude that the population variance from sample B is less than the population variance from sample A?

Homework Two samples from normally distributed populations yield the following summaries: (a) At 10% significance level, is there evidence that the population variance of A is greater than the population variance of B? (b) What assumptions do you make here about the two populations to justify the use of the F test? Sample A Sample B Mean 24.3 28.7 Variance 45.1 51.2 Sample size 21 11