P-VA LU E S

8.3 TESTING A PROPORTION

Testing a Proportion

• To test a proportion, we will use a similar process to those for testing the mean (8.1 & 8.2)• The main difference is that we are working with a

distribution of proportions

• Throughout this section we will assume that the situations we are dealing with satisfy the conditions of a binomial distribution• is the number of successes out of independent trials• is the estimate for , the population probability of success

on each trial• represents the probability of failure on each trial• The samples are larger: and • For large samples, the distribution is approximated by

the normal curve with and

Testing a Proportion

Left-Tailed Test

P-value

Right-Tailed Test

P-value

Two-Tailed Test

P-value

Hypotheses for Testing Note: p is a probability, therefore k must be between 0 and 1

1. Check requirements• Is this a binomial experiment with trials?• Does represent the probability of success?• Identify • Is the sample large? (Is and ?) (Use p from )• If yes, then the distribution can be approximated by the normal

distribution

2. State the null and alternate hypotheses, Identify the level of significance

3. Compute a sample test statistic• Standardized Sample Test Statistic:

4. Use the sampling distribution of the test statistic and the type of test to compute the P-value• use the standard normal distribution

5. Conclude the test• If P-value , we reject • If P-value > , we fail to reject

6. Interpret the results in the context of the application

How to Test

EXAMPLE: Testing

A team of eye surgeons has developed a new technique for a risky eye operation to restore the sight of people blinded from a certain disease. Under the old method, it is known that only 30% of the patients who undergo this operation recover their eyesight. Suppose that surgeons in various hospitals have performed a total of 225 operations using the new method and that 88 have been successful (i.e., the patients fully recovered their sight). Can we justify the claim that the new method is better than the old one? (Use a 1% level of significance.)

1. Check requirements• Is this a binomial experiment with trials? Yes, • Does represent the probability of success? Yes, • Identify = probability that patients recover their sight =

88• Is the sample large? (Is and ?) (Use p from )

Yes, 225(.30)=67.5 225(.70)=157.5• If yes, then the distribution can be approximated by the

normal distribution

2. State the null and alternate hypotheses, Identify the level of significance

EXAMPLE: Testing SOLUTION

3. Compute a sample test statistic

4. Use the sampling distribution of the test statistic and the type of test to compute the P-value

EXAMPLE: Testing SOLUTION

P-Value = Area of Shaded Region =Normalcdf(2.95,E99) .0016

5. Conclude the test• The P-value of .0016 of .01, therefore we reject

6. Interpret the results in the context of the application• At the 1% level of significance, the evidence shows the

population probability of success for the new surgery technique is higher than that of the old technique

EXAMPLE: Testing SOLUTION