Question

Data for the IFSAC Firefighter I examination test scores for two separate groups are found below. A random sample of firefighters is selected for each group. Group 1 attended the State Fire Academy for their training and Group 2 attended an in-house academy. The groups are only tested once after they have received the training. All participants have no prior experience in the fire service. Assume normality of the populations.

Group 1	Group 2
99	55
89	72
80	83
91	55
79	69
61	80
54	65
54	69
52	84
66	87
50	91
72	61
64	96
89	77
53	73
83	66
100	99

1.Verify that assumptions are met (briefly list and explain)

2.Construct the hypotheses (null and alternative)

3.Formulate decision rule (calculate the p-value or critical value)

4.Calculate the test statistic

5.Discuss your conclusion

Is there sufficient evidence that firefighters attending the in-house academy have higher test scores on average than firefighters attending the State Fire Academy at the α=.05 significance level?

Is there sufficient evidence that the mean scores on the IFSAC Firefighter I examination test differ between the two groups at the α=.05 significance level?

Answer 1

1.Verify that assumptions are met (briefly list and explain)

We are given that the populations are normal.

We assume that the population variances are equal.

2.Construct the hypotheses (null and alternative)

Null hypothesis: H₀: Firefighters attending the in-house academy have same test scores as firefighters attending the State Fire Academy.

Alternative hypothesis: H_a: Firefighters attending the in-house academy have higher test scores on average than firefighters attending the State Fire Academy.

3.Formulate decision rule (calculate the p-value or critical value)

We are given n1=17, n2=17

df = n1 +n2 – 2 = 17 + 17 – 2 = 32

α = 0.05

so, by using t-table

Critical value = 1.6939

Reject H₀ if test statistic value > 1.6939

4.Calculate the test statistic

Test statistic formula for pooled variance t test is given as below:

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

We are given

X1bar = 75.41176

X2bar =72.70588

S1 =13.34662

S2 = 17.30522

n1=17, n2=17

S_p² = [(17 – 1)* 17.30522^2 + (17 – 1)* 13.34662^2]/(17 + 17 – 2)

S_p² = 238.8015

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

t = (75.41176 - 72.70588) / sqrt[238.8015*((1/17)+(1/17))]

t = 2.7059/ sqrt[238.8015*((1/17)+(1/17))]

t = 2.7059/5.3004

t = 0.5105

5.Discuss your conclusion

Critical value = 1.6939

Test statistic = t = 0.5105

Test statistic < Critical value

So, we do not reject the null hypothesis

There is sufficient evidence to conclude that the mean scores on the IFSAC Firefighter I examination test is same for the two groups.