Question

9. The average number of cars per household in the year 2000 is known to have been 1.97. A researcher believes that the average has increased by the year 2018. In 2018 he surveys 155 households, and finds that for this sample, the mean is 2.05 with a standard deviation of .58 I would like you to test the researcher's belief statistically at the a = .05 level of significance. a. [3 pts) State the Null and Alternative hypotheses (in symbols and words) b. [4 pts) Perform the appropriate hypothesis test and produce relevant values (i.e., P- value and test statistics) c. [3 pts) Write a statement describing the result of your analysis and the conclusion. d. [3 pts) What happens if you test the hypothesis at the a=.01 level of significance i.e., Does the conclusion change? Why or why not?

Answer 1

a. The null and alternative hypothesis is ,

HO: 1.97

$H_a:\mu>1.97$

The test is right-tailed test.

b) The test statistic is ,

$t_{stat}=\frac{\bar{X}-\mu_0}{s/\sqrt{n}}=\frac{2.05-1.97}{0.58/\sqrt{155}}=1.717$

Now , df=degrees of freedom=n-1=155-1=154

The p-value is ,

p-value=$P(t_{df}>|t_{stat}|)=P(t_{154}>1.717)=0.044$ ; The Excel function is , =TDIST(1.717,154,1)

c) Decision : Here , p-value=0.044 < 0.05

Therefore , reject Ho.

Conclusion : There is sufficient evidence to support the claim that the average is increased by the year 2018.

d) Since , the significance level is 0.01

Decision : Here , p-value=0.044 > 0.01

Therefore , fail to reject Ho.

Conclusion : There is not sufficient evidence to support the claim that the average is increased by the year 2018.