Question

10.The distribution of vehicles sales in the United States in the year 2000 was the following:

Type of car Small Midsize Large Luxury

% 32.8 44.8 9.4 3.0

A simple random sample of 500 vehicles sales was taken this year and the Distribution is as follow:

Type of car Small Midsize Large Luxury

% 133 249 47 71

At the 5% significant level, do the data provides sufficient evidence to conclude that the distribution of vehicle sales this year has changed from the 2000 distribution?

Answer 1

Chi-Square Goodness of Fit test

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0: p1 =0.328, p2 =0.448, p3​=0.094, p4​=0.13 Ha​: 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) Degrees of Freedom The number of degrees of freedom is df=n-1=4-1=3 (3) Test Statistics The Chi-Squared statistic is computed as follows: (4)Critical Value and Rejection Region Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df=n-1=4-1=3, so the critical value becomes 7.8147. Then the rejection region for this test is R={χ2:χ2>7.8147}. (5)P-value The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 7.8147. The p-value is p=Pr(χ2>9.2038)=0.0267 (6) The decision about the null hypothesis Since it is observed that χ2=9.2038>χ2_crit​=7.8147, it is then concluded that the null hypothesis is rejected. (7) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the α=0.05 significance level.

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