Chapter 24: Comparing Means (when groups are independent) AP Statistics Sampling Distribution for the Difference of Two Means (when groups are independent) Sampling Distribution for the Difference of Two Means (when groups are independent) Formula for degrees of freedom when comparing means of independent groups The calculator will compute this for you Assumptions and Conditions Independence Assumption: Randomization Condition 10% Condition Normal Population Assumption: Need to check each group for normality. SHOW GRAPH. Nearly Normal Condition Independent Groups Assumption Just check for reasonability (this is very important) Two-Sample t-interval Two-Sample t-test Example Below are the saturated fat content (in grams) for several pizzas sold by two national chains. Create a 95% confidence interval for the difference in the means for the saturated fat content of the two brands. Brand D 17 12 1 0 8 8 10 10 5 16 16 8 12 15 7 11 11 13 13 11 12 Brand PJ 6 7 11 9 4 4 7 9 11 3 4 5 8 5 5 Example In order to create a twosample t-test, I first need to satisfy the Independent Sample Assumption, the Normal Population Assumption and the Independent Group Assumption. To satisfy these, I will need to satisfy the following conditions Example To satisfy the Independent Samples Assumption, we need to satisfy the below: Randomization Condition: We can assume that the pizzas from each company were picked at random 10% Condition: We assume that the 20 and 15 pizzas are both less than 10% of the pizzas made by each company Example To satisfy the Normal Population Condition, I can satisfy the Nearly Normal Condition (remember how sample size plays a role in what we look for) Brand D Brand PJ Both distributions of saturated fat roughly unimodal and symmetric. Example To satisfy the Independent Groups Assumption, I can assume that the groups are independent. There is no reason to think that the fat content in Brand D is not independent from the fat content in Brand PJ. Since all the Assumptions and Conditions have been met, we can use a t-distribution with 32.757 degrees of freedom and create a twosample t-interval. Example nD 20 y D 11.25 nPJ 15 y D 6.53 y D y D 4.72 sD 3.193 sPJ 2.588 df 32.757 Example SE y D y PJ 2 D 2 PJ s s nD nPJ 2 2 3.193 2.588 0.978 20 15 Example y D y PJ t * 32.8 SE y D y PJ 4.72 2.030.978 4.72 1.99 2.73,6.71 Example We are 95% confident that the true mean fat content of Brand D is between 2.73 and 6.71 grams higher than the true mean fat content for Brand PJ. Example Do the pizza chains have significantly different mean saturated fat contents? Conduct a hypothesis test.