Question

3. The Mathematics & Statistics department at Lancaster University claims that, on average, 25% of students obtain a first class degree. A competitor university is interested in showing that this value is an overestimate. They randomly sample 82 graduates and find that 18 have a host class degree

1. What are the assumptions for conducting a hypothesis test around this data? Are these satisfied?
2. What is your null hypothesis?
3. What is your alternative hypothesis?
4. Calculate the p-value for the hypotheses you proposed in (b) &(c).
5. What is your conclusion when conducting a 95% hypothesis test?
6. Do you agree with the competitor university that the proportion of first-class degrees advertised is an overestimate? State your reasoning.

Answer 1

This is problem of Hypothesis Test for a Proportion

a) To conduct a hypothesis test of a proportion, the following conditions are should be met:

• The sampling method is simple random sampling.
• Each sample point can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The sample includes at least 10 successes and 10 failures.
• The population size is at least 20 times as big as the sample size

In this context we have,

(sample size)

(no. of success)

First we need to check our assumptions that both np ≥ 10 and n(1−p) ≥ 10

np = 82 × 0.2195 = 17.999 and n(1−p)= 64.001 .Both are greater than 10 so this assumption has been met and we can use the standard normal approximation with this data.

b) Null Hypotheses

H_o: p = 0.25

c) Alternative Hypotheses

H_a: p < 0.25

This corresponds to a left-tailed test, for which a z-test for one population proportion needs to be used.

The z-statistic is computed as follows:

d) We need to find the corresponding level of p from the z value obtained. For this purpose, we need to look at the z table.

The p-value is p=0.2619

e) Since p=0.2619≥0.05, it is concluded that the null hypothesis is not rejected at 95% significant level.

f) NO, we are not agree with the competitor university that the proportion of first-class degrees advertised is an overestimate.