Independent Samples: Comparing Means Lecture 37 Sections 11.1 – 11.2, 11.4 Fri, Apr 6, 2007

The t Distribution Whenever we do not know , we must use s1 and s2 to estimate . In this case, we will have to use the t distribution instead of the standard normal distribution, unless the sample sizes are large.

Estimating  Individually, s1 and s2 estimate . However, we can get a better estimate than either one if we “pool” them together. The pooled estimate is

x1 –x2 and the t Distribution If we use sp instead of , and the sample sizes are small, then we should use t instead of Z. The number of degrees of freedom is df = df1 + df2 = n1 + n2 – 2. That is

Hypothesis Testing See Example 11.4, p. 699 – Comparing Two Headache Treatments. State the hypotheses. H0: 1 = 2 H1: 1 > 2 State the level of significance.  = 0.05.

The t Statistic Compute the value of the test statistic. The test statistic is with df = n1 + n2 – 2.

Computations

Hypothesis Testing Calculate the p-value. The number of degrees of freedom is df = df1 + df2 = 18. p-value = P(t > 1.416) = tcdf(1.416, E99, 18) = 0.0869.

Hypothesis Testing State the decision. State the conclusion. Accept H0. State the conclusion. Treatment 1 is more effective than Treatment 2.

The TI-83 and Means of Independent Samples – Stats Press STAT > TESTS. Choose 2-SampTTest. Choose Stats.

The TI-83 and Means of Independent Samples – Stats Provide the information that is called for. x1, s1, n1. x2, s2, n2. Alternative hypothesis. Whether to use a pooled estimate of . Answer “yes.”

The TI-83 and Means of Independent Samples – Stats Select Calculate and press ENTER. The display shows, among other things, the value of the test statistic and the p-value.

The TI-83 and Means of Independent Samples – Data Enter the data from the first sample into L1. Enter the data from the second sample into L2. Press STAT > TESTS. Choose 2-SampTTest. Choose Data.

The TI-83 and Means of Independent Samples – Data Provide the information that is called for. List 1: L1. List 2: L2. Freq 1: 1. Freq 2: 1. Alternative hypothesis. Whether to use a pooled estimate of . Answer “yes.”

The TI-83 and Means of Independent Samples – Data Select Calculate and press ENTER. The display shows, among other things, the value of the test statistic and the p-value.

Paired vs. Independent Samples The following data represent students’ calculus test scores before and after taking an algebra refresher course. Student 1 2 3 4 5 6 7 8 Before 85 63 94 78 75 82 45 58 After 92 68 98 83 80 88 53 62

Paired vs. Independent Samples Perform a test of the hypotheses H0: 2 – 1 = 0 H1: 2 – 1 > 0 treating the samples as independent.

Paired vs. Independent Samples Had we performed a test of the “same” hypotheses H0: D = 0 H1: D > 0 treating the samples as paired, then the p-value would have been 0.000005688. Why so small?

Paired Samples Why is there a difference? 1 2 3 5 4 6 8 7 40 50 60 80 90 100 70 Paired

Independent Samples Why is there a difference? 1 2 3 5 4 6 8 7 40 50 60 80 90 100 70 Independent

Confidence Intervals Confidence intervals for 1 – 2 use the same theory. The point estimate isx1 –x2. The standard deviation ofx1 –x2 is approximately

Confidence Intervals The confidence interval is or ( known, large samples) ( unknown, large samples) ( unknown, normal pops., small samples)

Confidence Intervals The choice depends on Whether  is known. Whether the populations are normal. Whether the sample sizes are large.

Example Find a 95% confidence interval for 1 – 2 in Example 11.4, p. 699. x1 –x2 = 3.2. sp = 5.052. Use t = 2.101. The confidence interval is 3.2  (2.101)(2.259) = 3.2  4.75.

The TI-83 and Means of Independent Samples To find a confidence interval for the difference between means on the TI-83, Press STAT > TESTS. Choose either 2-SampZInt or 2-SampTInt. Choose Data or Stats. Provide the information that is called for. 2-SampTTest will ask whether to use a pooled estimate of . Answer “yes.”