Question

A sports writer for the LA Times asserts that the proportion of games won by the Lakers over the past 40 years is 72%. You want to test if the proportion of won games may be different than the value the writer has claimed. You gather the results for a random sample of 80 games played by the Lakers within the past 40 years. Please actually solve all parts.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) What is the distribution of ˆp, the sample proportion?

(e) In your sample of 80 games, you find that the Lakers have won 55 games. You decide to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that the sports writer was incorrect?

(f) What is the p-value? What is the meaning of this number?

(g) For what values of the sample proportion would the null hypothesis be rejected?

(h) Calculate the probability of type II error if the true proportion is 78%.

(i) Solve (e), (g) and (h) when the level of significance is 0.01. Is your new answer for (e) consistent with the p-value found in (f)? How is the probability of type II error affected when the probability of type I error changes?

Answer 1

(a)

Hypotheses are:

(b)

The type I error:

The researcher incorrectly conclude that the proportion of won games is different from 0.72 while actually it is 0.72.

(c)

The type II error:

The researcher incorrectly conclude that the proportion of won games is not different from 0.72 while actually it is different from 0.72.

(d)

Here we have

p = 0.72, n=80

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

(e)

(f)

The p-value is 0.0008

It shows that under the assumption that null hypothesis is true probability of getting sample results is 0.0008.