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AP Statistics

AP Statistics. 13.1 Test for Goodness of Fit. Learning Objective:. Perform and analyze a chi-squared test for goodness of fit.

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AP Statistics

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  1. AP Statistics 13.1 Test for Goodness of Fit

  2. Learning Objective: • Perform and analyze a chi-squared test for goodness of fit.

  3. Chi-Squared ( ) test for goodness of fit- a single test that can be applied to see if the observed sample distribution is different from the hypothesized population distribution. • H₀:The distribution of the sample data is the SAME as the population • Ha:The distribution of the sample data is DIFFERENT than the population

  4. Test Name: Chi-Squared (Goodness of Fit) Test Chi-Squared test statistic:

  5. Assumptions: • Random Sample • All expected counts are greater than 1 • No more than 20% of expected counts are less than 5 • Degrees of freedom = n-1 where, n= # of categories

  6. Steps on your calculator (to calculate your p-value) • 1- L₁: (observed); L₂: (expected); L₃: (L₁- L₂)²/ L₂ • 2- sum(L ₃)= x² • 3- x²cdf(x²,1000,df)

  7. Properties of Chi-squared distribution: • Area under the curve= 1 • Skewed to the right • As the degrees of freedom increase, the closer to a normal distribution your curve becomes

  8. Ex 1: The “graying of America” is the recent belief that with better medicine and healthier lifestyles, people are living longer, and consequently a larger percentage of the population is of retirement age. Is this perception accurate? (Is there evidence that the distribution of ages changed drastically in 1996 from 1980?) • US Population by age group, 1980 Age Group Population(in thousandths) Percent 0-24 93,777 41.39 25-44 62,716 27.68 45-64 44,503 19.64 65-older 25,550 11.28 Total 226,546 100.00

  9. We select an SRS of 500. We first calculate our expected counts

  10. H₀: the distribution of the ages in 1996 is the SAME as it was in 1980 • Ha: the distribution of the ages in 1996 is DIFFERENT than it was in 1980 • Assumptions: -random sample -all expected counts are ≥ 1 -no more than 20% of expected counts are < 5 (See chart of expected counts) • Chi-squared test (goodness of fit) w/ α=0.05 • P(x²>8.214)= 0.042) df=4-1=3 • Since p<α, it is statistically significant, therefore we reject H₀. There is enough evidence to say the distribution of ages in 1996 is different than it was in 1980

  11. Ex 2: A wheel at a carnival game is divided into 5 equal parts. You suspect the wheel is unbalanced. The results of 100 spins are listed below. Perform a goodness of fit test. Is there evidence the wheel is not balanced? • H₀:The wheel is balanced • Ha: The wheel is unbalanced

  12. Assumptions: -random sample -all expected counts are ≥ 1 -no more than 20% of expected counts are < 5 Chi-Squared Test (goodness of fit) w/ α=0.05 P(x²>6.8)=0.146 df=4 Since p∡α, it is not statistically significant, therefore we do not reject H₀. There is not enough evidence to say the wheel is unbalanced.

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