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Testing Distributions

Testing Distributions. Section 13.1.2. Starter 13.1.2.

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Testing Distributions

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  1. Testing Distributions Section 13.1.2

  2. Starter 13.1.2 Elite distance runners are thinner than the rest of us. Skinfold thickness, which indirectly measures body fat, can show this. A random sample of 20 runners had a mean skinfold of 7.1 mm with a standard deviation of 1.0 mm. A random sample of 95 non-runners had a mean of 20.6 w/ sd of 9.0. Form a 95% confidence interval for the mean difference in body fat between runners and non-runners.

  3. Today’s Objectives • Students will graph the chi-square distributions with various degrees of freedom • Students will calculate the chi-square statistic for a distribution sample • Students will form hypotheses and determine whether a distribution is the same as or differs from a given distribution California Standard 19.0 Students are familiar with the chi- square distributions and chi- square test and understand their uses.

  4. The Big Picture (so far) • Chapter 11: Estimating means • One population: μ = some constant • Two populations: μ1 = μ2 • Chapter 12: Estimating proportions • One population: p = some percent • Two populations: p1 = p2 • Chapter 13: Comparing distributions • Distributions are categorical variables with more than 2 outcomes • Is a sample significantly different from population?

  5. The chi-square distributions • A family of distributions that take on only positive values • Curves are skewed right • A specific curve is determined by its degrees of freedom

  6. Shapes of the Distribution • Turn off all StatPlots, clear all Y= equations • Set window to [0, 14]1 x [-.05, .3].1 • Find the X2pdf distribution on the TI • Enter Y1 = X2pdf(x, 3) and draw the graph • Sketch the graph on a set of axes. • Label it X23 • Graph Y2 = X2pdf(x, 4) and add to your sketch. • Graph Y3 = X2pdf(x, 8) and add to your sketch.

  7. Properties of the chi-square distribution • The area under the curve is 1 • Each curve begins at x = 0, increases to a peak, then approaches the x axis asymptotically • Each curve is skewed to the right. As degrees of freedom increases, the curve becomes more symmetric and less skewed • Note that it can never become approximately normal: the graph is not symmetric

  8. The chi-square statistic • We will test the hypotheses by forming the chi-square statistic • The formula is: • Where “O” is the observed count and “E” is the expected count • We will ask the probability that X2 is as great as or greater than the value we found assuming the null hypothesis is true

  9. Assumptions needed for X2 test • Data are from a valid SRS from the population. • All individual expected counts are at least 1 • No more than 20% of the expected counts are less than 5

  10. Procedure One population t for means Two population t for means One population z for proportions Two population z for proportions X2 test for distributions Assumptions SRS, Normal Dist SRS, Normal, Independent SRS, large pop, 10 succ/fail SRS, large pop, 5 s/f each SRS, expect each count at least 1 No more than 20% less than 5 Assumptions So Far…

  11. Example • Here is the age distribution of Americans in 1980. Put the percents in L1 (as decimals)

  12. Example Continued • Here is a sample of 500 randomly selected individuals. Put the counts in L2

  13. Example Continued Now let’s find the X2 statistic: • For each age group, calculate the count you would expect if the 1980 distribution is true • Enter 500 x L1 into L3 (500 was the sample size) • For each age group, subtract the expected value from the actual count (observed) • Enter L2 – L3 into L4 • For each age group, square the (O-E) values and divide by E • Enter (L4)² / L3 into L5 • Add all the terms to get X2 • Sum(L5) • You should have gotten X² = 8.214

  14. The “Goodness of Fit” Test • Here are the hypotheses for distributions: • Ho: The distributions are the same • Ha: The distributions are different • To test the hypotheses, look at X2 • If the distributions are the same, then (O-E)=0, so X2 will be zero (or very low) • If the distributions differ, then (O-E)²>0, so X2 will be greater than the low number in case 1 • The more the distributions differ, the greater will be X2, the less will be the area under chi-square distribution to the right of X2

  15. Example Continued • Form hypotheses for the age distributions • Ho: The age distributions are the same • Ha: The age distributions differ • Find the X2 statistic • We already found X2 = 8.214 • Find the area to the right of X2 using n-1 d.f. • In this context, n is the number of categories • Enter Table E (p 842) at row 3 7.81<X2<9.35 so .05>p>.025 • Use the calculator X2cdf(8.214, 999, 3) = .0418 • Conclusion: There is good evidence (p<.05) to support the claim that the age distributions are different

  16. Today’s Objectives • Students will graph the chi-square distributions with various degrees of freedom • Students will calculate the chi-square statistic for a distribution sample • Students will form hypotheses and determine whether a distribution is the same as or differs from a given distribution California Standard 19.0 Students are familiar with the chi- square distributions and chi- square test and understand their uses.

  17. Homework • Read pages 702 – 709 • Do problems 1 - 4

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