Question

The U.S. Census Bureau computed the proportion of U.S residents who live in each of the four geographic regions in 2010. Then a simple random sample was drawn of 1000 people living in the U.S. in 2018. The following table presents the results:

Region	2010 Proportion	2018 Frequency
Northeast	0.179	173
Midwest	0.217	205
South	0.371	384
West	0.233	238

Determine whether the proportions of people living in the U.S. in the various regions has changed between 2010 and 2018. Use a 0.05 level of significance.

Prompt 1: Null Hypothesis

Prompt 1 Answer

Prompt 2: Number of degrees of freedom?

Prompt 2 Answer

Prompt 3: P-value

Prompt 3 Answer

Prompt 4Has the proportion of people living in the different regions of the U.S. changed from 2010 to 2018?

prompt 4: answer

Answer 1

H0: The proportion of people living in different region in 2010 and 2018 is same.

H0: the proportion of people living in different region has changed between 2010 and 2018.

Categories Observed Expected Northeast 173 1000*0.179=179 Midwest 205 1000*0.217=217 South 384 1000*0.371=371 West 238 1000*0.233=233 Sum = 1000 1000

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H. : P1 = 0.179, P2 = 0.217, p3 = 0.371, P4 = 0.233 He: Some of the population proportions differ from the values stated in the null hypothesis This corresponds to a Chi-Square test for Goodness of Fit. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, the number of degrees of freedom is df = 4–1 = 3, so then the rejection region for this test is R= {x? : x2 > 7.815}. :

(3) Test Statistics The Chi-Squared statistic is computed as follows: n (Oi x2 = Σ E;)? Ei - 0.201 + 0.664 +0.456 + 0.107 = 1.428 i=1 (4) Decision about the null hypothesis Since it is observed that x² = 1.428 < x2 = 7.815, it is then concluded that the null hypothesis is not rejected.

(5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is NOT enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the a = 0.05 significance level.
thus it may concluded that the proportion of people living in different region has not changed over 2010 to 2018.

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