1.6. Tests of the population proportion (Large sample)

Often we want to conduct test of hypothesis about a population proportion.

This section presents the procedure to perform tests of hypothesis about the population proportion. p for large samples ( . The procedure to make such tests is similar in many respects to the one for the population mean .

The value of the test statistic for the sample proportion computed as

where -is the sample proportion, and the value of used in this formula is the one used in the null hypothesis.

Then, if the number of sample observations is large and observed proportion is , the following tests have significance level :

1. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or

.

Once again, is the number for which

and is the standard normal distribution.

Example:

Mr. A and Mr. B are running for local public office in a large city. Mr. A says that only 30% of the voters are in favor of a certain issue, a law to sell liquor on Sundays. Mr. B doubts A’s statement and believes that more than 30% favor such legislation. Mr. B pays for an independent organization to make a study of this situation. In a random sample 400 voters, 160 favored the legislation. What conclusions should the polling organization report to Mr. B?

Solution:

Let be proportion of all people who favor such legislation and the corresponding sample proportion. Then from given information,

; ; . Let .

The null and alternative hypotheses are as follows

The decision rule is to reject the null hypothesis in favor of alternative if

; .

.

and

From the given information we calculate the value of test statistic as

Since we reject . We make conclusion that more than 30% of voters are in favor of a law to sell liquor on Sundays.

Exercises

1. Make the following hypothesis tests about p.

a) ; ; ; ;

b) ; ; ; ;

c) ; ; ; ;

2. Consider versus .

a) A random sample of 600 observations produced a sample proportion equal to 0.67. Using , would you reject the null hypothesis?

b) Another random sample of 600 observations taken from the same population produced a sample proportion of 0.76. Using , would you reject the null hypothesis?

Comment on the result of parts a) and b).

3. A food company is planning to market a new type of ice cream. Before marketing this ice cream, the company wants to find what percentage of the people like it. The company’s management has decided that it will market this ice cream only if at least 35% of people like it. The company’s research department selected a random sample of 400 persons and asked them to test this ice cream. Of these 400 persons, 128 said they liked it.

a) Testing at 2.5% significance level, can you conclude that the company should market this yogurt?

b) What will your decision be in part a) if the probability of making a Type I error is zero?

4. A mail order company claims that at least 60% of all orders are mailed within 48 hours. The quality control department took a sample of 500 orders and found that 310 of them were mailed within 48 hours of the placement of the orders. Testing at 1% significance level, can you conclude that the company’s claim is true?

5. Let p=proportion of adults in a city who required a lawyer in the past year.

a) Determine the rejection region for level test of

against .

b) If 65 persons in a random sample of 200 required lawyer services, what does the test conclude?

6. A magazine claims that 25% of its readers are university students. A random sample of 200 readers is taken and 42 of these readers are university students. Use level of significance to test the validity of the magazine’s claim.

7. Suppose that in order to test the hypothesis that against the alternative that ,we decide to obtain a sample of size 100 and reject if we obtain fewer than 48 successes.

a) What is the approximate size of the Type I error?

b) If the value of p is really 0.5, what is the size of Type II error?

8. An educator wishes to test against , where

p-proportion of football players who graduate university in four years.

a) State the test statistic and the rejection region having .

b) If 19 out of a random sample of 48 players graduated in four years, what does the test conclude? Also evaluate p-value.

9. The president of a company that produces national brand coffee claims that 40% of the people prefer to buy national brand coffee. A random sample of 700 people who buy coffee showed that 252 of them buy national brand coffee.

a) Using , can you conclude that the percentage of people who buy national brand coffee is different from 40%?

b) Find the p-value for the test. Using this p-value, would you reject the null hypothesis at ? What if ?

Answers

1. a) ; do not reject ; b) ; reject ;

c) ; do not reject ; 2. a) ; do not reject ;

b) ; reject ; 3. a) ; do not reject ; b) do not reject ; 4. accept ; 5. a) ; b) ; reject ; 6. ; accept ; 7. a) about 0.0071 b)approximately 0.6554; 8. a) ; b) ; accept for ;

9. a) ; do not reject ;b) p-value=0.0308; reject at ; do not reject at