In: Statistics and Probability
The marital status distribution of the U.S. male population, ages 15 and older, is as follows: 31.3% never married, 56.1% married, 2.5% widowed, and 10.1% divorced/separated. A random sample of 400 U.S. young adult males, 18 to 24 years old, found 140 never married, 238 married, 2 widowed, and 20 divorced/separated. Using α = 0.10, is this evidence that males in this age group follow a different distribution than all males in the U.S.? Write the hypotheses, calculate the expected counts, check the condition, calculate the test statistic, and use either the critical value approach or the p-value approach to make a conclusion about the question being asked.
We need to check whether males in this age group follow a different distribution than all males in the US. We will perform a chi-square goodness of fit test.
Ho: The observed distribution of males does not significantly differ from the expected distribution i.e Oi = Ei
Ha: The observed distribution of males significantly differs from the expected distribution i.e Oi =/ Ei
The expected counts are calculated as follows:
Oi | Ei (%) | Expected counts (%age*400) | |
Never Married | 140 | 31.30% | 125.2 |
Married | 238 | 56.10% | 224.4 |
Widowed | 2 | 2.50% | 10 |
Divorced/separated | 20 | 10.10% | 40.4 |
400 |
As all the expected counts > 5, the condition is checked and we can continue to perform the analysis.
The test statistic is calculated by:
The calculations are:
Oi | Ei (%) | Ei (%age*400) | (Oi-Ei) | (Oi-Ei)*(Oi-Ei) | (Oi-Ei)*(Oi-Ei)/Ei | |
Never Married | 140 | 31.30% | 125.2 | 14.8 | 219.04 | 1.75 |
Married | 238 | 56.10% | 224.4 | 13.6 | 184.96 | 0.82 |
Widowed | 2 | 2.50% | 10 | -8 | 64 | 6.4 |
Divorced/separated | 20 | 10.10% | 40.4 | -20.4 | 416.16 | 10.3 |
400 | Total | 19.27 |
Chi-square value = 19.275
The critical chi-square value at df = 4 -1 = 3 and alpha = 0.1 is: 6.251
As calculated chi-square = 19.275 > 6.251, it falls in the rejection region. Hence, we reject the Ho. We conclude that the males in this age group follow a different distribution than all the males in the US.