Question

In a random sample of 35 Penn State Shenango students, 17 were attending full-time. In a random sample of 40 World Campus students, 13 were attending full-time.. Use the five-step hypothesis testing procedure given below to determine if there is evidence of a difference between the proportions of all Penn State Shenango and all Penn State World Campus students who are attending full-time.

Step 1: Check assumptions and write hypotheses

Step 2: Calculate the test statistic

̩Step 3: Determine the p-value

Step 4: Decide between the null and alternative hypotheses

Step 5: State a real world conclusion

Answer 1

Step 1:

The test procedure used in this context is  two-proportion z-test,following are the assumptions for this test

• The sampling method for each population is simple random sampling.
• The samples are independent.
• Each sample includes at least 10 successes and 10 failures.

Null and alternative hypothesis

H_o : p₁ = p₂ v/s H₁ : p₁ ≠ p₂

Step 2 :

We have

p₁ = 17/35 =0.4857

p₂ = 13/40 = 0.325

The pooled sample proportion is given by

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

= (0.4857*35 + 0.325*40)/ (35 +40)

= 0.4

Standard error

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }

= sqrt{ 0.4 * ( 1 - 0.4 ) * [ (1/35) + (1/40) ] }

=

= 0.1134

The test statistic in this case is given by

z = (p₁ - p₂) / SE

= (0.4857 - 0.325)/0.1134

= 1.417

Step 3 : P -value

Using  the Normal Distribution Calculator or table, the probability associated with the z-score is

P-value = 0.0782

Step 4 : Since p-value is less than greater than 0.05, we are unable to reject the null hypothesis.

Hence null hypothesis is accepted.

Step 5 :

There is no evidence of a difference between the proportions of all Penn State Shenango and all Penn State World Campus students who are attending full-time.