Question

Suppose the National Transportation Safety Board (NTSB) wants to examine the safety of compact cars, midsize cars, and full-size cars. It collects a sample of cars each of the cars types. The data below displays the frontal crash test performance percentages.

Compact Cars	Midsize Cars	Full-Size Cars
95	95	93
98	98	97
87	98	92
99	89	92
99	94	84
94	88	87
99	93	88
98	99	89

Patrick wants to purchase a new car, but he is concerned about safety ratings. Using the data from the chart above, what would you recommend to Patrick if he is debating between compact, midsize, and full-size cars? FYI: High scores on crash performance tests are GOOD. (Higher scores means they passed the test a higher percent of the time.)

1. Evaluate all three types of car in your response using One-Way ANOVA and follow-up t-tests. 2. Explain why you gave him this suggestion.

Answer 1

The result of One-Way ANOVA is shown below(Calculated using MS.Excel):

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Compact Cars	8	769	96.125	17.26786
Midsize Cars	8	754	94.25	17.07143
Full-Size Cars	8	722	90.25	16.5
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	144.0833	2	72.04167	4.251142	0.028172	3.4668
Within Groups	355.875	21	16.94643
Total	499.9583	23

The Null hypothesis here states that there is no difference between the mean of the three different types of cars(i.e., compact cars, midsize cars, and full-size cars) in the crash test performance percentages. From the One-Way ANOVA result, it is seen that the p-value is less than 0.05, which indicates that there is no evidence to accept the null hypothesis. Hence the One-Way ANOVA result states that there is a significant difference among the different types of cars.

We can see that the average crash test performance percentages are higher for Compact Cars and Midsize Cars. Therefore, we should be suggesting either Compact Cars and Midsize Cars by checking whether there is any significant difference among them using the two-sample t-test. The t-test is carried out using MS Excel and the result is shown below.

t-Test: Two-Sample Assuming Equal Variances
	Compact Cars	Midsize Cars
Mean	96.125	94.25
Variance	17.26785714	17.07142857
Observations	8	8
Pooled Variance	17.16964286
Hypothesized Mean Difference	0
df	14
t Stat	0.905004288
P(T<=t) one-tail	0.190386932
t Critical one-tail	1.761310136
P(T<=t) two-tail	0.380773865
t Critical two-tail	2.144786688

Here the p-value (consider P(T<=t) two-tail) is greater than 0.05. Hence the null hypothesis is rejected, clearly indicates that the average crash test performance percentages is highest in Compact Cars.

So, from the analysis, it is clear that Patrick should go for Compact Cars as it has the highest average crash test performance percentages as compared to midsize cars, and full-size cars.