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1 10.1 The Sampling Distribution of the Difference Between Two Sample Means for Independent Samples Tom Lewis Fall Term The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for1 Indepe / 6
2 Outline 1 The rationale 2 A small example 3 Normal populations 10.1 The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for2 Indepe / 6
3 The rationale A typical problem Do women do better on the SAT than men? How could we test for this? 10.1 The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for3 Indepe / 6
4 The rationale A typical problem Do women do better on the SAT than men? How could we test for this? There are two populations under consideration: the men and the women The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for3 Indepe / 6
5 The rationale A typical problem Do women do better on the SAT than men? How could we test for this? There are two populations under consideration: the men and the women. There is a common statistic under consideration: the SAT score The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for3 Indepe / 6
6 The rationale A typical problem Do women do better on the SAT than men? How could we test for this? There are two populations under consideration: the men and the women. There is a common statistic under consideration: the SAT score. Each population has its own population mean SAT score: µ 1 for the boys and µ 2 for the girls The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for3 Indepe / 6
7 The rationale A typical problem Do women do better on the SAT than men? How could we test for this? There are two populations under consideration: the men and the women. There is a common statistic under consideration: the SAT score. Each population has its own population mean SAT score: µ 1 for the boys and µ 2 for the girls. We can collect random samples from each population and compute the sample means of their SAT scores: x 1 for the boys and x 2 for the girls The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for3 Indepe / 6
8 The rationale A typical problem Do women do better on the SAT than men? How could we test for this? There are two populations under consideration: the men and the women. There is a common statistic under consideration: the SAT score. Each population has its own population mean SAT score: µ 1 for the boys and µ 2 for the girls. We can collect random samples from each population and compute the sample means of their SAT scores: x 1 for the boys and x 2 for the girls. How can we compare the sample means? How much of a difference between the sample means, x 2 x 1, is sufficient to assert that there is a difference in the population means, µ 2 µ The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for3 Indepe / 6
9 A small example A small example Here are the scores on a recent exam for a group of boys and girls: Alex Bob Chuck Denise Ellen Fergie Gisele The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for4 Indepe / 6
10 A small example A small example Here are the scores on a recent exam for a group of boys and girls: Alex Bob Chuck Denise Ellen Fergie Gisele Find all samples of size 2 from the boys and all samples of size three from the girls. Find the values of the mean of the scores for each of the random samples. Let x 1 be the mean of the boy s samples and let x 2 denote the means of the girl s samples The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for4 Indepe / 6
11 A small example A small example Here are the scores on a recent exam for a group of boys and girls: Alex Bob Chuck Denise Ellen Fergie Gisele Find all samples of size 2 from the boys and all samples of size three from the girls. Find the values of the mean of the scores for each of the random samples. Let x 1 be the mean of the boy s samples and let x 2 denote the means of the girl s samples. Find all 12 possible values of x 1 x The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for4 Indepe / 6
12 A small example A small example Here are the scores on a recent exam for a group of boys and girls: Alex Bob Chuck Denise Ellen Fergie Gisele Find all samples of size 2 from the boys and all samples of size three from the girls. Find the values of the mean of the scores for each of the random samples. Let x 1 be the mean of the boy s samples and let x 2 denote the means of the girl s samples. Find all 12 possible values of x 1 x 2. Find the mean and standard deviation of the values of x 1 x The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for4 Indepe / 6
13 Normal populations Normal data Our next result is significant, but it requires that the variable under question be normally distributed within the two populations The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for5 Indepe / 6
14 Normal populations Normal data Our next result is significant, but it requires that the variable under question be normally distributed within the two populations. Theorem Suppose that x is a normally distributed variable on each of two populations. Then, for independent samples of size n 1 and n 2 from the two populations, 10.1 The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for5 Indepe / 6
15 Normal populations Normal data Our next result is significant, but it requires that the variable under question be normally distributed within the two populations. Theorem Suppose that x is a normally distributed variable on each of two populations. Then, for independent samples of size n 1 and n 2 from the two populations, µ x1 x 2 = µ 1 µ 2, 10.1 The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for5 Indepe / 6
16 Normal populations Normal data Our next result is significant, but it requires that the variable under question be normally distributed within the two populations. Theorem Suppose that x is a normally distributed variable on each of two populations. Then, for independent samples of size n 1 and n 2 from the two populations, µ x1 x 2 = µ 1 µ 2, σ x1 x 2 = (σ1 2/n 1) + (σ2 2/n 2) 10.1 The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for5 Indepe / 6
17 Normal populations Normal data Our next result is significant, but it requires that the variable under question be normally distributed within the two populations. Theorem Suppose that x is a normally distributed variable on each of two populations. Then, for independent samples of size n 1 and n 2 from the two populations, µ x1 x 2 = µ 1 µ 2, σ x1 x 2 = (σ1 2/n 1) + (σ2 2/n 2) x 1 x 2 is normally distributed The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for5 Indepe / 6
18 Normal populations Problem Work problems and from the text The Sampling Distribution of the Difference Between Two FallSample Term 2009 Means for6 Indepe / 6
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