1. AP Statistics Chapter 13 Section 1

2. 2 kinds of Chi – Squared tests 1.Chi-square goodness of fit – extends inference on proportions to more than 2 proportions by enabling us to determine if a particular population distribution has changed from a specified form. 2.Chi-square test of independence – further extends this notion by allowing us to test whether or not the distribution of one variable has been influenced by another variable.

3. Chi-Square Test 1.Null hypothesis – written in words 2.Alternative hypothesis – written in words – always “different” 3.Alpha level 4.appropriate bar graph 5.Find Counts and Expected Counts 6.Assumptions 7.Find Chi-Square Test statistic 8.Find P-value 9.Write Conclusion 10.If in the conclusion, the null hypothesis is rejected – follow up analysis

4. “The graying of America”? There is a belief that due to better medicine and healthier lifestyles, people are living longer, and consequently a larger percentage of the population is of retirement age. Is this accurate?

5. Age Group Population (in thousands) Percent 0 to 2493,77741.39 25 to 4462,71627.68 45 to 6444,50319.64 > 6525,55011.28 Total226,546100.00 U.S. population by age group, 1980 Age Group CountPercent 0 to 2417735.4 25 to 4415831.6 45 to 6410120.2 > 656412.8 Total500100.00 Sample results for 500 randomly selected individuals, 1996 Is this evidence that America is in fact “graying”?

6. The largest difference lies in the 0 to 24 age group.

7. The age group distribution in 1996 is the same as the 1980 distribution. The age group distribution in 1996 is different from the 1980 distribution. The idea of the test: Compare observed counts with the counts that would be expected if the 1996 distribution were the same as the 1980 distribution. The 1980 distribution is the population. The more the observed counts differ from the expected counts, the more evidence to reject the null hypothesis.

8. Assumptions for Chi-square 1. All individual expected counts are at least 1. 2.No more than 20% of the expected counts are less than 5.

9. Age Group1980 population percents Expected counts 0 to 2441.39500(.4139)=207 25 to 4427.68500(.2768)=138.4 45 to 6419.64500(.1964)=98.2 > 6511.28500(.1128)=56.4 Total100500

10. Age GroupObservedExpected(O-E) 2 E 0 to 241772074.3478 25 to 44158138.42.7757 45 to 6410198.2.0798 > 656456.41.0241 Total8.2275

11. Assumptions 1.All expected counts are > 1. 207, 138.4, 98.2, 56.4 2. No expected counts are < 5. Chart on pg. 842

12. Conclusion Given that the two distributions are the same the observed difference between the expected values and the observed values from the 1996 random sample would occur less than 5 out of every 100 times just by chance. Therefore, this evidence rejects that the two distributions are the same. The largest difference falls into the youngest age category (0 to 24). There is a small difference in the oldest category but it does not represent the largest difference.

13. Calculator 1.For the Chi-square goodness of fit test – do the math 2.The calculator will be helpful for the Chi- square test of independence in 13.2.