Question

A researcher at the Nie Pójdzie Motor Club is interested in the average repair bill for automobile owners when the “check engine” light comes on. He believes that automobiles manufactured overseas (Import) will have lower repair costs (in dollars) than automobiles manufactured in the United States (Domestic). A random sample from each type of manufacturer was drawn.   

The Data Analysis output for the various tests used when comparing two group means are shown below. The significance level was .05.   Identify which two sample test should be performed from the scenario and from the following outputs. Then, answer the questions (a to f) using the correct t test output. (3.5 pts.)

F-Test Two-Sample for Variances
	Import	Domestic
Mean	331.2	349.6
Variance	569.07	104.27
Observations	10	10
df	9	9
F	5.4578
P(F<=f) one-tail	0.0094
F Critical one-tail	3.1789

t-Test: Two-Sample Assuming Equal Variances
	Import	Domestic
Mean	331.2	349.6
Variance	569.07	104.27
Observations	10	10
Pooled Variance	336.67
Hypothesized Mean Difference	0
df	18
t Stat	-2.2423
P(T<=t) one-tail	0.0189
t Critical one-tail	1.7341
P(T<=t) two-tail	0.0378
t Critical two-tail	2.1009

t-Test: Two-Sample Assuming Unequal Variances
	Import	Domestic
Mean	331.2	349.6
Variance	569.07	104.27
Observations	10	10
Hypothesized Mean Difference	0
df	12
t Stat	-2.242
P(T<=t) one-tail	0.0223
t Critical one-tail	1.7823
P(T<=t) two-tail	0.0446
t Critical two-tail	2.1788

t-Test: Paired Two Sample for Means
	Import	Domestic
Mean	331.2	349.6
Variance	569.07	104.27
Observations	10	10
Pearson Correlation	1.00
Hypothesized Mean Difference	0
df	9
t Stat	-4.263
P(T<=t) one-tail	0.0011
t Critical one-tail	1.8331
P(T<=t) two-tail	0.0021
t Critical two-tail	2.2622

a) What is the appropriate two sample test to perform – the paired t test, the t test assuming equal variances, or the t test assuming unequal variances – for this research project?

b) State the H₀ and H_a.

c) Identify the decision rule using the critical value of t (round to three decimal places).

d) Identify the decision rule using the p value method.

e) State the test statistic (t _calc).

f) Do you reject or not reject Ho? Explain your decision.

Answer 1

Concept

If we have to find whether there is a difference between the two groups within a population or within two population, we use three types of t-tests

1. Two sample t-test with Equal variances
2. Two sample t-test Unequal variances
3. Paired sample t test

In general, population variances are not known, so we use pooled variance with t-distribution instead of normal distribution. In case of paired sample test, two values from same samples are taken, the two samples are not independent of each other

a)

• In this problem, we have to compare the mean repair costs in the USA to that of outside USA. So the samples are independent and hence we can reject the paired sample t-test.
• Now to find whether we have to use equal variance test or unequal variance test, we can take help from the first table, F-test for sample variance.
• F-test is based on the hypothesis
• Ho: Variances are equal.
• Ha: Variances are not equal
• From the 1st table, we see that the p-value is 0.0094 which is less than 0.05 (alpha). So we can reject the null hypothesis, and conclude that variances are not equal.

So, the appropriate test to perform is using two-sample t-test for unequal variances

b)

Hypothesis are

c)

Decision Rule using Critical value of t

Reject Ho if t-value < t-critical

t-value = -2.242 (taken from table 3, one tail is we have inequality sign in alternate hypothesis)

t-critical = -1.782 ( as we have to take the left tail probablity )

Since, t-value < t-critical . Reject Null

d)

Decision Rule using p-value

Reject Ho if p-value < alpha (significance level, 0.05)

p-value = 0.0223 (taken from table 3, one tail)

Since, p-value < 0.05 . Reject Null

e)

Test-statistic can be taken directly from table 3, for one tail as the hypothesis has an inequality (<) sign. We take a two-tailed probability when we have an " unequal" sign in the alternate hypothesis.

So, test statistic = -2.242

f)

As we have seen that both the decision rule, p-value and t-statistic suggest that we have to reject the null hypothesis.

We reject null and conclude that the automobile manufactured overseas will have lower repair cost than that of USA