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. Test whether there are statistical differences in the frontal crash test performance for each type of car.

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

What conclusions can we draw from the follow-up t-tests?  

There is/are a total of ["1", "2", "3", "4", "5", "6"] statistically significant difference(s) between car type pairs out of the follow-up t-tests.

Answer 1

i) To test Null hypothesis

against Alternative hypothesis H₁ : not all means are equal

Using Excel, (Data -> Data Analysis -> Anova: Single Factor), we get the following output -

Here,

The value of test statistic F = 4.25114

and critical value F_critical = 3.4668

Since F statistic > F_critical , so we reject the null hypothesis at 5% level of significance and we can conclude that there is significant differences in the frontal crash test performance for each type of car.

ii) 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.