The US census bureau computed the proportion of US residents who lived in each of 4 geographic regions in Year A. Then a simple random sample was drawn of 1000 people living in the US in year b. The following data is the results. Northeast: Year A :0.181, Year B: 152; Midwest: year a : 0.248, year b :227; South Year A: 0.343, year b 390; West: year a: 0.228, year b: 231;

Question

Can you conclude that the proportion of people living in the various regions changed between year a and year b. Use the a= 0.05 level of significance. First find the P- value. Round to four decimal places

Answer 1

To determine if there was a significant change in the proportion of people living in the various regions between Year A and Year B, we can conduct a chi-squared goodness of fit test.

Hypotheses:
Null Hypothesis (H0): The proportions of people living in the various regions did not change between Year A and Year B.
Alternative Hypothesis (Ha): The proportions of people living in the various regions changed between Year A and Year B.

First, calculate the expected frequencies for each region in Year B based on the proportions from Year A.
Northeast: 152 * 0.181 = 27.512
Midwest: 227 * 0.248 = 56.296
South: 390 * 0.343 = 133.770
West: 231 * 0.228 = 52.668

Now, calculate the chi-squared test statistic:
chi-squared = Σ((observed - expected)^2 / expected)
= ((152-27.512)^2 / 27.512) + ((227-56.296)^2 / 56.296) + ((390-133.770)^2 / 133.770) + ((231-52.668)^2 / 52.668)
= 435.861

Degrees of freedom = 4 - 1 = 3

Next, find the p-value associated with the chi-squared test statistic of 435.861 and 3 degrees of freedom. Since the p-value is essentially zero, we reject the null hypothesis.

Conclusion:
Based on the p-value of essentially zero, we can conclude that there was a significant change in the proportions of people living in the various regions between Year A and Year B.