Chapter 7.2-7.5 - 002

Ch7.2-7.5-page 1 7.2 Paired data Paired t-C.I., paired t-test Data format (n pairs of data from n subjects): Subject A B C D E Before (x) X 1 X 2 X 3 X 4 X 5 After (y) Y 1 Y 2 Y 3 Y 4 Y 5 X diff = x-y Let D µ = ____________________________________________________________ Confidence Interval for µ D : * diff diff df S x t n ± Test statistic : 0 diff D diff x t S n µ − = has t n-1 if n >30 or X diff is nearly normally distributed

Ch7.2-7.5-page 2 Example 1. Are textbooks actually cheaper online? Below are 6 textbooks’ prices in the UCLA bookstore and on Amazon. Book1 B2 B3 B4 B5 B6 UCLA (x) 27.67 40.59 31.68 16 23.76 27.70 Amazon (y) 27.95 31.14 32 11.52 28.74 18.22 a) Test the appropriate hypothesis to answer the question. b) Compute and interpret a 95% CI for the mean price difference between UCLA and Amazon.

Ch7.2-7.5-page 4 Step 4 Conclude H 0 and write a sentence to answer the question. b) Construct a 95% confidence interval for the true mean difference in Before and After . c) Conclude the hypotheses set up in part a) based on the confidence interval obtained above.

Ch7.2-7.5-page 5 7.3 Difference of two means 2 sample t-C.I., 2 sample t-test Conditions : Two independent samples (NOT Paired!) that are either nearly normally distributed, or both samples are large enough (≥ 30) Sample info : Sample 1: n 1 , 1 x , S 1 Sample 2: n 2 , 2 x , S 2 Parameters : µ 1 = true mean of population 1, µ 2 = true mean of population 2 Wanted : inference about (µ 1 - µ 2 ): Method s: 2 sample t-CI, 2 sample t-test CI for (µ 1 - µ 2 ) : ( ) 2 2 * 1 2 1 2 1 2 df S S x x t n n − ± + , df =smaller(n 1 -1, n 2 -1) test stat 1 2 0 2 2 1 2 1 2 x x t s s n n − = + has a t-distribution with degrees of freedom of df =smaller(n 1 -1, n 2 -1) Example 1. Do male and female newborn babies have the same weight on average? 2 independent samples: Male: n1 = 10, 1 x = 7 lb 11 oz = 123 oz, s 1 = 8 oz Female: n2 = 8, 2 x = 7 lb 4 oz = 116 oz, s 2 = 5 oz Each sample is nearly normally distributed. a) Compute and interpret a 95% CI for the mean difference in weight between male and female newborn babies. Then, answer the question based on the CI. b) Perform a hypothesis test to answer the question

Ch7.2-7.5-page 7 Class EX1. A random sample of 50 people in Canada convicted of murder charges had an average sentence length of 22.6 years and standard deviation of 2.9 years. A random sample of 75 people in the USA convicted of murder charges had an average sentence length of 23.1 years with a standard deviation of 2.7 years. At α =0.05 is there significant evidence to suggest that the average prison sentence is longer in the US than in Canada? Answer the question by completing the following parts: a) Define parameter to be tested and formulate hypotheses b) Check the conditions and calculate the statistic c) Find the p-value d) Draw a conclusion regarding the null hypothesis and answer the question in a sentence e) Calculate and interpret a 95% confidence interval for the difference of the two true mean prison sentence lengths.

Ch7.2-7.5-page 8 7.5 Comparing many means with ANOVA Example 1. An experiment was designed to examine whether an infant’s attention span varied with the types of designs on baby mobiles. A group of 30 infants was randomly divided into 5 equal groups. Each group was shown a mobile with one of 5 multicolored designs: A, B, C, D, or E. The median time that the infants spent looking at the design was recorded in the spreadsheet. Perform a one-way ANOVA to test whether the 5 designs have equal mean time ? Data: Analysis of Variance (ANOVA) MegaStat output : H 0 : H 1 : Test statistic F = P-value = Conclusion : Assumptions for ANOVA: