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Data and Analysis

This analysis focuses on understanding the differences between men's and women's cholesterol levels through rigorous hypothesis testing and regression analysis. We calculate the standard error of the difference and assess its significance relative to the observed difference in cholesterol levels. Additionally, we explore how income affects health expenditures, demonstrating that higher income correlates with increased spending. The analysis emphasizes the importance of clear hypothesis formulation, appropriate sample selection, and precise calculations of central tendencies and dispersions.

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Data and Analysis

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  1. Data and Analysis

  2. Let’s Recall Some Numbers • Early on, we wanted to look at the difference between men and women … • w.r.t. cholesterol

  3. From EXCEL CH_3.XLS

  4. Hypothesis testing • In the book's example (sample of 30 each), which is based on real life numbers, we find that:

  5. Testing Differences • Let's see if we can do things a bit more rigorously, looking at the "spread" of the data. • So:

  6. Testing Differences • It turns out that we can calculate a standard error of the difference which equals: • sd = (sw2 + sm2)1/2 • sd= (4.5 + 4.2)1/2 = (19.36 + 16.81)1/2 = 6.17. • Let's compare the spread, or standard error, of the difference to the difference. The difference of 10.2 is about 1.65 times as big as the standard error. Clearly, if the difference was 0, we'd accept the fact that cw = cm.

  7. Testing Differences • The larger the difference relative to the spread, the more likely that cw cm. • If d/sd = 1.645, we could be about 90% certain that men's cholesterol is not equal to women's. • As it turns out, that’s just about the certainty that we get here.

  8. Key points for hypothesis testing: State hypotheses clearly Choose suitable sample Calculate appropriate measures of central tendency and dispersion Draw appropriate inferences

  9. Regression Analysis • Difference of means is useful, but sometimes we want to be a little more detailed. • Suppose that we wanted to know how people’s expenditures changed as their incomes went up. • One simple example is “rich people spend more than poor people.” • You get a sample of rich people, and then a sample of poor people. Calculate the mean for the rich people, and the mean for the poor people. • Hypothesis? Er > Ep!

  10. Example for Health Expenditures • Let’s collect some data. Expenditures • What does this suggest? • Rich people spend more. • How much more? • Let’s draw a line. Income per capita

  11. Example for Health Expenditures b = slope • Line has a form: Exp = a + b*income Expenditures • What does a mean? • What does b mean? a Income per capita

  12. Example for Health Expenditures b = slope • Says that for each $ of income per capita, we spend $b more. Expenditures • Although it is hard to think of, we could draw this diagram in n dimensions! • What else determines health expenditures? a Income per capita

  13. EXCEL Example

  14. Calculating Elasticities y = a + b x. Dy = bD x. Why? Dy/Dx= b. Is this an elasticity? Remember, the elasticity is in percentage terms, so: E = %Dy/ %Dx = (Dy/y)/(Dx/x). Can re-write this as: E = %Dy/ %Dx = (Dy/y)/(Dx/x) = (Dy/Dx)(x/y). Dy/Dx= b, so: E = b(x/y).

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