Question

A medical researcher wants to compare the pulse rates of smokers and non-smokers. He believes that the pulse rate for smokers and non-smokers is different and wants to test this claim at the 0.02  level of significance. The researcher checks 79 smokers and finds that they have a mean pulse rate of 87, and 75 non-smokers have a mean pulse rate of 83. The standard deviation of the pulse rates is found to be 6 for smokers and 6 for non-smokers. Let μ1 be the true mean pulse rate for smokers and μ2 be the true mean pulse rate for non-smokers.

Step 1 of 4 :

State the null and alternative hypotheses for the test.

Step 2 of 4:

Compute the value of the test statistic. Round your answer to two decimal places.

Step 3 of 4:

Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to three decimal places.

Step 4 of 4:

Make the decision for the hypothesis test.

Answer 1

Solution:

Here, we have to use two sample t test for difference between two population means by assuming equal population variances.

Null hypothesis: H0: The pulse rate for smokers and non-smokers is same.

Alternative hypothesis: Ha: The pulse rate for smokers and non-smokers is different.

H0: µ1 = µ2 versus Ha: µ1 ≠ µ2

This is a two tailed test.

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

t = (X1bar – X2bar) / sqrt[Sp2*((1/n1)+(1/n2))]

Where Sp2 is pooled variance

Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

We are given

X1bar = 87

X2bar = 83

S1 = 6

S2 = 6

n1 = 79

n2 = 75

df = n1 + n2 – 2 = 79 + 75 – 2 =

α = 0.02

Critical value = - 2.3511 and 2.3511

(by using t-table)

Sp2 = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

Sp2 = [(79 – 1)*6^2 + (75 – 1)*6^2]/(79 + 75 – 2)

Sp2 =36

t = (X1bar – X2bar) / sqrt[Sp2*((1/n1)+(1/n2))]

t = (87 – 83) / sqrt[36*((1/79)+(1/75))]

t = 4/0.9673

t = 4.1352

Test statistic = t = 4.1352

P-value = 0.0001

(by using t-table or excel)

P-value < α = 0.02

So, we reject the null hypothesis

There is sufficient evidence to conclude that the pulse rate for smokers and non-smokers is different.