Testing the Difference Between Means Independent Samples 1

Testing the Difference Between Means (Independent Samples, 1 and 2 Known) .

Two Sample Hypothesis Test • • . Compares two parameters from two populations. Sampling methods: § Independent Samples • The sample selected from one population is not related to the sample selected from the second population. § Dependent Samples (paired or matched samples) • Each member of one sample corresponds to a member of the other sample.

Independent and Dependent Samples . Independent Samples Dependent Samples Sample 1 Sample 2

Example: Independent and Dependent Samples Classify the pair of samples as independent or dependent. • Sample 1: Weights of 65 college students before their freshman year begins. • Sample 2: Weights of the same 65 college students after their freshmen year. Solution: Dependent Samples (The samples can be paired with respect to each student).

Example: Independent and Dependent Samples Classify the pair of samples as independent or dependent. • Sample 1: Scores for 38 adults males on a psycho- logical screening test for attention-deficit hyperactivity disorder. • Sample 2: Scores for 50 adult females on a psycho- logical screening test for attention-deficit hyperactivity disorder. Solution: Independent Samples (Not possible to form a pairing between the members of the samples; the sample sizes are different, and the data represent scores for different individuals. ).

Two Sample Hypothesis Test with Independent Samples 1. Null hypothesis H 0 § A statistical hypothesis that usually states there is no difference between the parameters of two populations. § Always contains the symbol , =, or . 2. Alternative hypothesis Ha § A statistical hypothesis that is true when H 0 is false. § Always contains the symbol >, , or <. .

Two Sample Hypothesis Test with Independent Samples H 0: μ 1 = μ 2 Ha : μ 1 ≠ μ 2 H 0: μ 1 ≤ μ 2 Ha : μ 1 > μ 2 H 0: μ 1 ≥ μ 2 Ha : μ 1 < μ 2 Regardless of which hypotheses you use, you always assume there is no difference between the population means, or μ 1 = μ 2. .

Two Sample z-Test for the Difference Between Means Three conditions are necessary to perform a z-test for the difference between two population means μ 1 and μ 2. 1. The population standard deviations are known. 2. The samples are randomly selected. 3. The samples are independent. 4. The populations are normally distributed or each sample size is at least 30. .

Two Sample z-Test for the Difference Between Means • Test statistic is • The standardized test statistic is Can be used when these conditions are met: 1. Both σ1 and σ2 are known. 2. The samples are random. 3. The samples are independent. 4. The populations are normally distributed or both n 1 ≥ 30 and n 2 ≥ 30. .

Example: Two-Sample z-Test for the Difference Between Means A credit card watchdog group claims that there is a difference in the mean credit card debts of households in California and Illinois. The results of a random survey of 250 households from each state are shown at the left. The two samples are independent. Assume that σ1 = $1045 for California and σ2 = $1350 for Illinois. Do the results support the group’s claim? Use α = 0. 05. (Source: Plastic. Economy. com) .

Solution: Two-Sample z-Test for the Difference Between Means • • • . H 0: μ 1 = μ 2 H a: μ 1 ≠ μ 2 0. 05 n 1= 250 , n 2 = 250 Rejection Region: • Test Statistic: • Decision: Fail to Reject H 0 At the 5% level of significance, there is not enough evidence to support the group’s claim that there is a difference in the mean credit card debts of households in New York and Texas.

Example: Using Technology to Perform a Two-Sample z-Test A travel agency claims that the average daily cost of meals and lodging for vacationing in Texas is less than the average daily cost in Virginia. The table at the left shows the results of a random survey of vacationers in each state. The two samples are independent. Assume that σ1 = $19 for Texas and σ2 = $24 for Virginia, and that both populations are normally distributed. At α = 0. 01, is there enough evidence to support the claim? (Adapted from American Automobile Association) .

Solution: Using Technology to Perform a Two-Sample z-Test • Rejection Region: 0. 01 -2. 33 0 -0. 91 . z • Decision: Fail to Reject H 0 At the 1% level of significance, there is not enough evidence to support the travel agency’s claim.

Testing the Difference Between Means (Independent Samples, 1 and 2 Unknown) .

Two Sample t-Test for the Difference Between Means A two-sample t-test is used to test the difference between two population means μ 1 and μ 2 when 1. σ2 and σ2 are unknown, 2. the samples are random, 3. the samples are independent, and 4. the populations are normally distributed or both n 1 ≥ 30 and n 2 ≥ 30. .

Two Sample t-Test for the Difference Between Means • The standardized test statistic is • The standard error and the degrees of freedom of the sampling distribution depend on whether the population variances and are equal. .

Two Sample t-Test for the Difference Between Means • Variances are equal (pooled) § Information from the two samples is combined to calculate a pooled estimate of the standard deviation § The standard error for the sampling distribution of is § d. f. = n 1 + n 2 – 2.

Two Sample t-Test for the Difference Between Means • Variances are not equal (not pooled) § If the population variances are not equal, then the standard error is § d. f = smaller of n 1 – 1 or n 2 – 1 .

Normal (z) or t-Distribution? .

Example: Two-Sample t-Test for the Difference Between Means The results of a state mathematics test for random samples of students taught by two different teachers at the same school are shown below. Can you conclude that there is a difference in the mean mathematics test scores for the students of the two teachers? Use α = 0. 10. Assume the populations are normally distributed and the population variances are not equal. . Teacher 1 Teacher 2 s 1 = 39. 7 s 2 = 24. 5 n 1 = 8 n 2 = 18

Solution: Two-Sample t-Test for the Difference Between Means • • • . H 0: μ 1 = μ 2 H a: μ 1 ≠ μ 2 0. 10 d. f. = 8 – 1 = 7 Rejection Region: • Test Statistic: • Decision: Fail to Reject H 0 At the 10% level of significance, there is not enough evidence to support the claim that the mean mathematics test scores for the students of the two teachers are different.

Example: Two-Sample t-Test for the Difference Between Means 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 similar 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 (1) Competition (2) s 1 = 45 ft s 2 = 30 ft n 1 = 14 n 2 = 16

Solution: Two-Sample t-Test for the Difference Between Means • • • H 0: μ 1 ≤ μ 2 H a: μ 1 > μ 2 0. 05 d. f. = 14 + 16 – 2 = 28 Rejection Region: 0. 05 0 . 1. 701 t • Test Statistic: • Decision:

Solution: Two-Sample t-Test for the Difference Between Means • • • H 0: μ 1 ≤ μ 2 H a: μ 1 > μ 2 0. 05 d. f. = 14 + 16 – 2 = 28 Rejection Region: 0. 05 0 1. 701 1. 811. t • Test Statistic: • Decision: Reject H 0 At the 5% level of significance, there is enough evidence to support the manufacturer’s claim that its phone has a greater calling range than its competitors.