2 Independent Samples for Means Statistics Lesson 22: Individual Preparation Name: Before we find a confidence interval or conduct a hypothesis test involving the means of two independent samples, the following requirements MUST be met: I The two samples are independent. I Both samples were randomly selected. I Both samples come from normally distributed populations OR n1 2 30 and n2 2 30. Note: As we very rarely know sigma for the two populations that we want to work with, we will also assume 01 and 02 are both unknown and no assumption about their equality is made. Thus, we will only by working with t distributions for this lesson. Hypothesis Test Involving the Means of Two Independent Samples 1. Two different methods of teaching Italian vocabulary were tried on two independent groups. Method 1 was used with a group of 40 students selected at random. Method 2 was used on another group of 42 students selected at random. After one month, the same vocabulary test was given to both groups. The average score (out of 100) for the group using method 1 was 87 with standard deviation 4.0. The average score for the group using method 2 was 89 with standard deviation 4.2. Test the claim that students taught with method 2 perform better on the vocabulary test. Use a 1% level of significance. a. Requirement Check {Remember to make sure BOTH samples meet each requirement!) b. Write down known information and hypotheses. Population #1 1 = Population it 2 = Ho: #1 = #2 0R PitP12 = 0 H1: ,ul #2 0R pl \"2 0 (in blanks place , or #5 based on problem) Type of test: - tailed a = c. Find the critical values. Then draw and label the picture of the distribution. d.f.= the smallerof {1111 OR n21}= tc= (flf2) 512 + i "'1 \"2 Then label the test statistic on the picture. d. Find the test statistic t = e. Use the test statistic to find the p-value. f. Using the test statistic and the p-value, state your conclusion. g. Now let's use yourTl-83/84 calculator to find the test statistic and pvalue. Go to STAT -> TEST -> 4: 2-SampTTest (beca use there are ; samples, the critical value 8:. test statistic are tscores, and we are doing a hypothesis te_st.). Input the required information (and use the \"STATS\" option because you don't have the original data set.) Write down what you see in the space below. Confidence Interval Involving the Means of Two Independent Samples 2. Sam is com paring the sugar content of ice cream and yogurt. A random sample of 10 servings of yogurt had an average of 14.6 g and a standard deviation of 2.9 g. An independent random sample of 12 servings of ice cream had an average sugar content of 22.4 g with a standard deviation of 3.2 g. Given that the sugar content for each treat is normally distributed, find (and interpret) a 90% confidence interval for the difference of the two population means. a. Requirement Check Write down the known information. Population it 1 = Population it 2 = Pu Th H H- {3" I'D In 5 Pi E "l O ._.., F'*"'\\ H: l H O :U :1 M l H \\_....J H H- n H Make the confidence interval (f1 2 ) E 0 and, therefore, #1 > #2. * If the ENTIRE interval is negative, then (.11 #2 TESTS -> 0:2-SampTlnt (beca use there are 2 samples, we are using :- scores for the critical values, and we are finding a confidence interval). Since wejust have the summary statistics rather than the actual sample data, choose the \"Stats\" option. Next input fl, 3x1, n1,f2,5x2, 112, and the confidence level. Choose "No\" for the Pooled option. Finally, select \"Calculate\". Write what you see below. Note: The interval from the calculator and the interval found using the table and d.f. formula is a bit different. This is because we used a simplified d.f. formula for the hand calculation, which resulted in a wider interval