Question

A researcher wished to see if there is evidence that graduates of one of the top business programs perfom better than other investment managers. To do this, she tested the hypothesis: Null hypothesis: p = 0.25 Alternative Hypothesis: P > 0.25 Using a random sample of size n=100. In her sample, she found that the proportion of graduates of the top business programs who performed better than other investment managers was 35%.

1. Calculate the standard deviation value of the Normal Distribution model that she can use to test the above null hypothesis.

2. Find the standard score of the sample proportion of graduates of the top business programs who performed better than other investment managers (show your calculation).

3. Calculate the p-value of the test, using Table A.

4. Sketch the p-value here (sketch a normal distribution, mark its mean and "shade" the probability that shows the p-value for this test).

5. Based on the calculated p-value, do we reject the null hypothesis or not? Explain your answer (use 5% significance level).

Answer 1

1. Calculate the standard deviation value of the Normal Distribution model that she can use to test the above null hypothesis.

Required standard deviation = sqrt(pq/n)

We are given

n = 100

p = 0.25

q = 1 – p = 0.75

Required standard deviation = sqrt(0.25*0.75/100)

Required standard deviation = 0.043301

2. Find the standard score of the sample proportion of graduates of the top business programs who performed better than other investment managers (show your calculation).

The required standard score Z is given as below:

Z = (p̂ - p)/sqrt(pq/n)

n = sample size = 100

p̂ = x/n = 0.35

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.35 – 0.25)/ sqrt(0.25*0.75/100)

Z = (0.35 – 0.25)/ 0.043301

Z = 2.3094

Required standard score = 2.3094

3. Calculate the p-value of the test, using Table A.

The p-value by using z-table is given as below:

P-value = 0.0105

4. Sketch the p-value here (sketch a normal distribution, mark its mean and "shade" the probability that shows the p-value for this test).

Required sketch of normal curve is given as below:

P-value 3 -2 1 0 1

5. Based on the calculated p-value, do we reject the null hypothesis or not? Explain your answer (use 5% significance level).

We have

P-value = 0.0105

α = 0.05

P-value < α

So, we reject the null hypothesis

There is sufficient evidence to conclude that the proportion of graduates of the top business programs who performed better than other investment managers is more than 25%.