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Chapter 24: Comparing Means (when groups are independent) AP Statistics
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Sampling Distribution for the Difference of Two Means (when groups are independent)
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Formula for degrees of freedom when comparing means of independent groups The calculator will compute this for you
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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)
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Two-Sample t-interval
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Two-Sample t-test
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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 D17 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
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Example In order to create a two- sample 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
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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
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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 DBrand PJ Both distributions of saturated fat roughly unimodal and symmetric.
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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 two- sample t-interval.
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Example
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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.
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Example Do the pizza chains have significantly different mean saturated fat contents? Conduct a hypothesis test.
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