Final answer:
To evaluate if the distribution of top safety picks by the Insurance Institute for Highway Safety remained the same between 2009 and 2013, a chi-square test for homogeneity is used. This test will indicate whether observed changes in the distribution of awards are statistically significant at the 5 percent level.
Step-by-step explanation:
When assessing the safety standards of automobiles, the Insurance Institute for Highway Safety plays a pivotal role by compiling annual safety reports. To determine whether the distribution of top safety picks has remained consistent from 2009 to 2013, we analyze data from Table 11.58 which categorizes top safety picks across six car categories. In approaching this problem, we will apply statistical methods to evaluate the data at the 5 percent significance level.
Statistical analysis is essential in determining whether changes in the distribution of top safety picks awards are random or indicative of an actual shift in vehicle safety standards. By conducting a chi-square test for homogeneity, we can compare the frequencies of top safety picks between the two years in question. If the P-value obtained from the chi-square test is less than the alpha level of 0.05, we reject the null hypothesis that the distribution of top safety picks has remained the same.
Given this statistical framework, we would examine the frequencies for each category of vehicle across both years, summing the chi-square values. Should the total chi-square statistic exceed the critical value for our degrees of freedom, it supports a change in the distribution of top safety picks. Conversely, if the statistic is less than the critical value, we fail to reject the null hypothesis, suggesting no significant change in safety awards distributions.
In summary, we must rely on professional statistical methods to arrive at a conclusion about the stability of the distribution of top safety picks between 2009 and 2013. This analysis serves not only to elucidate trends in automobile safety improvements but also informs consumer and manufacturer decisions. The significance or lack thereof determined by our chi-square test will yield insights into how safety priorities may have shifted over the observed period.