Question

Create a case on the application of inferences about the difference between two population means (σ1 and σ2 known) and explain the hypothesis tests until conclusion.

Answer 1

Introducing a case-

Two different types of drink (i) chocolate milk and (ii) carbohydrate replacement were given to two set of cyclists and after a rest they are asked to perform cycling until complete exhaustion and times (in minutes) are recorded.. From prior knowledge it is known that standard deviations are 12 minutes and 9 minutes respectively. Times of complete exhaustion are given in following table.

Chocolate milk	Carbohydrate replacement
44	48
26	31
37	30
58	25
52	42
26	40
58	28
40
48

Perform a statistical test to justify whether there is any difference in mean time of exhaustion for these two items or not at 10% level of significance.

ANSWER-

Suppose, random variable X denotes time of exhaustion (in minutes) in case of chocolate milk and random variable Y denotes time of exhaustion (in minutes) in case of carbohydrate replacement.

We have to test for null hypothesis

against the alternative hypothesis

Our test statistics is given by

Here,

First sample size

Second sample size

[Using R-code '1-pnorm(1.59309)+pnorm(-1.59309)']

Level of significance

We reject our null hypothesis if .

Here we observe that .

So we can not reject our null hypothesis.

Hence, based on given information we conclude that there is no significant difference between two population means.