Hypothesis Tests About a Proportion

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We now would like to test the following three null hypotheses about a population proportion p:

1. Ho: p <= P
2. Ho: p >= P
3. Ho: p = P


We can test each claim simultaneously with a sample proportion m / n, where m is the number of favorable (or "Yes") responses and n is the random sample size. If m / n is too large, then we must reject the first null hypothesis Ho: p <= P. If m / n is too small, then we reject the second null hypothesis. If m / n is either too large or too small, then we reject the third null hypothesis.

Once again, we conduct the tests with the use of the test statistic. If the population is considered "large," then we define the test statistic by

x = (m / n - P) / Sqrt[P(1 - P) / n].


If the population is of a smaller, finite size N (so that the sample size n is more than 5% of the entire population), then we define the test statistic by

x = (m / n - P) / [ Sqrt[ P(1 - P) / n] Sqrt[ (N - n) / (N - 1) ] ].


In each case, the test statistic follows an approximate standard normal distribution Z ~ N(0,1) for large sample sizes. The tail probabilities of this statistic are then computed using Z.

If a is our pre-chosen level of significance, then we reject the first null hypothesis if P(Z >= x) < a. This case is equivalent to the test Ho: p = P with a one-sided alternative Ha: p > P.

We reject the second null hypothesis if P(Z <= x) < a. This case is equivalent to the test Ho: p = P with a one sided alternative Ha: p < P.

We reject the third null hypothesis if P(Z <= x) < a / 2 or P(Z >= x) < a / 2.



Using the PTEST Program

The PTEST program can be used to perform these hypothesis tests. To execute the program, first enter 1 or 2 to designate a "large" population or a finite population of known size N which you then enter. Next, enter the value of the proportion to be tested, along with the number of "Yes" responses, the sample size, and the level of significance.

The program displays the conclusion for all three of these tests, along with the value of the test statistic and the left and right tail values.



Example. On June 30, 1995, The Associated Press reported the results of a poll commisioned by the Center on Addiction and Substance Abuse at Columbia University. The poll found that 304 of 400 youths interviewed believed that popular culture encourages drug use. Test the following three claims, at the 0.05 level of significance, about the true proportion p of youths that have this belief at that time.

1. Ho: p <= 0.70
2. Ho: p >= 0.70
3. Ho: p = 0.70


Solution. We first note that the sample proportion is 304 / 400 = 0.76; thus we will automatically accept the second hypothesis.

After calling up the PTEST program, first enter 1 to desigate a large population. Next enter .70 for TEST PROPORTION, enter 304 for NUMBER OF YES, enter 400 for SAMPLE SIZE, and enter .05 for LEVEL OF SIG.

With a right tail value of 0.0044, we find that we reject the first and third claims, but accept the second claim.

We can conclude that p > 0.70. If p <= 0.70 or if p = 0.70, then there would be at most a 0.44% chance of the sample proportion being as large as 0.76 with a random sample of size 400; thus, we reject both of these claims. (Here the p-value for the first hypothesis test is 0.0044, but the p-value for the two-sided test Ho: p = 0.70 is 2*.0044 = 0.0088.)



Exercises

1. In a recent national poll, 63% of 900 adults surveyed stated that they agreed with the "war on terror." Test the following three claims, at the 0.05 level of significance, about the true proportion p of adults that agree with the war on terror.

1. Ho: p <= 0.60
2. Ho: p >= 0.60
3. Ho: p = 0.60


2. In a college of 2650 students, 264 out of 400 surveyed had registered to vote. Test the following three claims, at the 0.06 level of significance, about the true proportion p of students who are registered to vote at this college.

1. Ho: p <= 0.70
2. Ho: p >= 0.70
3. Ho: p = 0.70


3. On June 25, 1995, The Associated Press reported the results of a national survey conducted by the Center for Social and Religious Research at the Hartford Seminary. The study was on the divorce rate of a group of 5000 Protestant clergymen and 5000 Protestent clergywomen. It was found that 25% out of 2458 clergywomen responding had been divorced at least once.

Test the following three claims, at the .03 level of significance, about the true proportion p of the entire 5000 clergywomen that had been divorced at least once.

1. Ho: p <= 0.225
2. Ho: p >= 0.225
3. Ho: p = 0.225


4. A decade-old study found that 60% of entering college freshmen believed that "getting rich" was an important personal goal. A researcher decides to test whether or not that percentage still stands. His new survey finds that 520 out of 840 college-bound high school seniors believe that "getting rich" is an important goal. State and perform a test that refutes or upholds the previous percentage.



Solutions

1. After calling up the PTEST program, first enter 1 to designate a large population, then enter .60 for TEST PROPORTION. Since the exact number of "Yes" responses is not given, we can enter .63*900 for NUMBER OF YES, then enter 900 for SAMPLE SIZE. Lastly enter .05 for LEVEL OF SIG.

We receive a right tail value of 0.0331 from a test statistic of1.837 and we have evidence, with a = 0.05, to reject the first hypothesis.

In particular, if p <= 0.60, then there would be at most a 3.31% chance of the sample proportion being as large as 0.63 from a random sample of 900 adults. This low p-value of 0.0331 gives us significant evidence to reject the first hypothesis.

However for the two-sided test, the p-value is 2*.0331 = 0.0662, which is above our designated level of significance; thus, we would not reject the third hypothesis. We ultimately conclude that p >= 0.60.


2. After calling up the PTEST program, enter 2 for a finite population, then enter the population size of 2650. Next, enter .70 for TEST PROPORTION, 264 for NUMBER OF YES, 400 for SAMPLE SIZE, and .06 for LEVEL OF SIG.

We receive a left tail value of 0.0291. At the 0.06 level of significance, we reject the second and third hypotheses, and thus accept the claim that p < 0.70.

We note that the sample proportion is 264 / 400 = 0.66. For the second test, if p >= 0.70, then there would be no more than a 2.81% chance of a sample proportion being as low as 0.66 with a random sample of size 400 from this population. Thus we reject this hypothesis since the p-value of 0.0291 is below 0.06.

For the two-sided test Ho: p = 0.70, the p-value of 2*.0291 = 0.0582 which is also below the level of significance; thus we again reject the hypothesis.


3. After calling up the program, first enter 2 for a finite population, then enter the population size of 5000. Next, enter .225 for TEST PROPORTION. Since the exact number of "Yes" responses is not given, we can enter .25*2458 for NUMBER OF YES and 2458 for SAMPLE SIZE. Lastly enter .03 for LEVEL OF SIG.

We receive a right tail value of 0 and we reject the first and third hypotheses.

Since we have accepted the second but rejected the third, we can conclude that p > 0.225. If p <= 0.225 or if p = 0.225, then there would be no chance of the sample proportion being as large as 0.25 with this many responses from this finite population. Thus, we reject these claims.


4. We shall test the null hypothesis Ho: p = 0.60 with a two-sided alternative. Executing the PTEST program for a "large" population with a 0.05 level of significance, we obtain a test statistic of 1.1269 and a right-tail value of 0.1299, which gives a two-sided p-value of 0.2598. With this large p-value, we do not have significant evidence to refute the claim that p = 0.60. If p were equal to 0.60, then there could still be a 12.99% chance of a sample proportion being as large as 520 / 840 = 0.619 from a random sample of size 840.



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