Question

5. The chi-square test for goodness of fit - No difference from a known population Aa Aa Suppose you are reading a study conducted in the year 2000 about welfare recipients in the United States. The authors report the following frequency data on the household size of the 2,352 welfare recipients in their random sample: Observed Frequencies Household Size 5-or-more-person 4-person 3-person 2-person 1-person 282 753 588 400 329 You wonder if welfare recipients tend to live in different-sized households than the U.S. population at large. You obtain the following data from the 2000 census: Percent Distribution of U.S. Households by Size Household Size 5-or-more-person 4-person 3-person 2-person 1-person 10.83% 14.20% 16.53% 32.63% 25.81% (Source: Hobbs F., & Stoops, N. (2002). Consus 2000 special reports: Domographic trends in the 20th century, U.S. Consus Bureau) You use a chi-square test for goodness of fit to see how well the sample of welfare recipients fits the census data, What is the most appropriate null hypothesis? The distribution of household sizes among welfare recipients is equal across the five household-size categories The distribution of household sizes among welfare recipients is not equal across the five household-size categories The distribution of household sizes among welfare recipients is different from that provided by the census data. The distribution of household sizes among welfare recipients is the same as that provided by the census data,

Answer 1

The distribution of household sizes among welfare recipients is the same as that provided by the census data,

applying chi square goodness of fit test:

	relative	observed	Expected	residual	Chi square
Category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
5 or more	0.1083	282	254.72	1.709	2.921
4	0.1420	400	333.98	3.612	13.049
3	0.1653	329	388.79	-3.032	9.194
2	0.3263	753	767.46	-0.522	0.272
1	0.2581	588	607.05	-0.773	0.598
total	1.00	2352	2352	0.994298552	26.03
test statistic X2=		26.034

expected values are in expected column

for 0.01 level and 4 df :crtiical value X2 =

13.277

from excel: chiinv(0.01,4)

can reject the null hypothesis. Because the chi-square statistic X2 is greater the critical value, you Therefore, you can conclude that the proportions of welfare recipients in the five household-size categories differ from the corresponding proportions for all U.S. households.