Section 9 2B:!! Using Confidence Intervals to Estimate the Difference ( µ 1 µ 2 ) in Two Population Means using Two Independent Samples.

Transcription

1 Section 9 2B:!! Using Confidence Intervals to Estimate the Difference ( µ 1 µ 2 ) in Two Population Means using Two Independent Samples Requirements 1.A random sample of each population is taken. The sample mean for the sample of one population is x 1 and the sample mean for the sample of a second population is x 2 2. The two samples are independent of each other. 3. Both populations are normal or both sample sizes are greater than 30. Notation for the Samples of Two Population Means! Population 1! Population 2 µ 1 = population mean! µ 2 = population mean n 1 = sample size! n 2 = sample size x 1 = Sample Mean from Population 1! x 2 = Sample Mean from Population 2 s 1 = Sample Standard Deviation! s 2 = Sample Standard Deviation from Population 1! from Population 2 Creating a Confidence Interval to Estimate the value of the Difference in 2 Population Means µ 1 µ 2 (x 1 x 2 ) E < (µ 1 µ 2 ) < (x 1 x 2 ) + E if x 1 > x 2 or (x 2 x 1 ) E < (µ 2 µ 1 ) < (x 2 x 1 ) + E if x 2 > x 1 Where E = t α 2 ( ) 2 s 1 ( ) 2 n 1 + s 2 n 2 Use the smallest Degrees of Freedom from Sample 1 and Sample 2 Note: It does not matter which population is population 1 and population 2. It is important that the difference between them is positive. The first formula is based on x 1 being larger than x 2. The second formula is based on x 2 being larger than x 1. You are free to chose which population is selected to be population. If you always chose the population with the largest x as population 1 then you can always use the first formula. Section 9 2B Lecture! Page 1 of 5! 2018 Eitel

2 Example 1 Is Ortho Grow fertilizer more effective in lawn growth then a generic fertilizer? Ortho Grow brand fertilizer selected 50 lawns and measured how much the grass had grown in 30 days. The average growth was 2.18 inches with a standard deviation of.64 inches. A generic fertilizer was also tested on 35 randomly selected lawn. The average growth for the generic brand was 2.37 inches with a standard deviation of.85 inches. Construct a 90% confidence interval for the difference between the two means. Does it appear there is a difference in the two population means? How can you tell? Sample 1!! Sample 2 The sample with the larger mean! The sample with the smaller mean mean is labeled Sample 1:! is labeled Sample 2:! x 1 = 2.37 s 1 =.85! n 1 = 35 x 1 = 2.18 s 1 =.64 n 1 = 50 Use the smallest Degrees of Freedom from Sample 1 and Sample 2.!! Confidence Interval ( x 1 x 2 ) E < µ 1 µ 2 < ( x 1 x 2 ) + E ( ).29 < µ 1 µ 2 < ( ) < µ 1 µ 2 <.48 Conclusion based on the problem: The confidence interval does contain zero. I am 90% confident that here is no difference in the average growth of the lawn using Ortho or the generic fertilizer. Section 9 2B Lecture! Page 2 of 5! 2018 Eitel

3 t Distribution: Critical t Values Degrees of Area In One Tail (Right Tail) Freedom Section 9 2B Lecture! Page 3 of 5! 2018 Eitel

4 Example 2 Does completing the homework in Statistics help improve your test score? Mr. Jensen selected 51 Statistics students at random from his classes who had completed all the Chapter 9 homework. The average score on the Chapter 9 test for these students was 92.5 points with a standard deviation of 2 points. Mr. Jensen selected 54 Statistics students at random from his classes who had not completed all the Chapter 9 homework. The average score on the Chapter 9 test for these students was 83 points with a standard deviation of 4 points. Construct a 95% confidence interval for the difference between the two means. Dos it appear there is a difference in the two population means? How can you tell? Sample 1!! Sample 2 The sample with the larger mean! The sample with the smaller mean mean is labeled Sample 1:! is labeled Sample 2:! x 1 = 92.5 s 1 = 2 x 1 = 83 s 1 = 4 n 1 = 51! n 1 = 54 Use the smallest Degrees of Freedom from Sample 1 and Sample 2.! Confidence Interval ( x 1 x 2 ) E < µ 1 µ 2 < ( x 1 x 2 ) + E ( ) 1.23 < µ 1 µ 2 < ( ) < µ 1 µ 2 <10.73 Conclusion based on the problem: The confidence does not contain zero. I am 95% confident that there is difference at the.05 significance level in the test scores for students that complete the chapter homework. They score from 8.3 to points higher on the test Section 9 2B Lecture! Page 4 of 5! 2018 Eitel

5 t Distribution: Critical t Values Degrees of Area In One Tail (Right Tail) Freedom Section 9 2B Lecture! Page 5 of 5! 2018 Eitel