9.10: Testing a Claim about Population Proportion

A complete procedure for testing a claim about a population proportion is provided here.

There are two methods of testing a claim about a population proportion: (1) Using the sample proportion from the data where a binomial distribution is approximated to the normal distribution and (2) Using the binomial probabilities calculated from the data.

The first method uses normal distribution as an approximation to the binomial distribution. The requirements are as follows: sample size is large enough, the probability of proportion p is close to 0.5, the np (product of sample size and proportion) is greater than 5, and the critical values can be calculated using the z distribution. It also requires the samples to be random and unbiased, and the nature of the data to be binomial, i.e., there are only two possible outcomes (e.g., success or failure; selected or not selected, true or false, etc.). A proportion is binomial in nature. So, this method is well suitable for testing a claim using hypothesis testing for population proportion.

As a first step, the hypotheses (null and alternative hypotheses) are stated clearly and expressed symbolically. The proportion p used in hypotheses statements is the assumed proportion value, often 0.5. The proportion obtained from the data is the sample proportion. Both these values are crucial in calculating the z statistic.

The critical value can then be obtained from the z distribution utilizing the normal approximation of the binomial distribution. The critical value can be positive or negative based on the hypothesis direction; accordingly, the hypothesis test is right-tailed, left-tailed, or two-tailed. The critical value is calculated at any desired confidence level, most commonly 95% or 99%.

The P-value is then directly calculated using the z statistic and the critical z value, and the hypothesis test is concluded. The z statistic can also be directly compared with the critical value to conclude the hypothesis test.

The second method of testing the claim about proportion does not require np > 5 as it uses the exact binomial distribution without normal approximation. This method does not calculate the critical value. Instead, it uses the probabilities of obtaining x (the value of successes out of total trials, e.g., 60 successes out of 110 trials) in the n trials. It calculates the probabilities of x or fewer and x or greater and then leads to the P-values. This second method of testing a claim about proportion is tedious to do manually and requires statistical software. Nonetheless, the inferences determined in both ways are equally accurate.

In the natural populations of Trinidadian guppies, females select the males with orange coloration for mating.

To determine if guppy populations in an aquarium also show the same behavior, an experiment is conducted where 12 females are individually introduced to three orange males and three blue males simultaneously.

It is originally claimed that the females choose the orange males.

So, the null hypothesis would state that an equal number of females would show a preference for orange and blue males. The alternative hypothesis is that a higher number of females would prefer the orange males.

The experiment shows that ten out of twelve females preferred the orange males.

This ratio provides the sample proportion— 0.83—which is used to get the test statistic as follows.

It is observed that this test statistic falls within the critical region at a significance level of 0.05.

Also, the P-value from this z statistic is 0.011.

So, we may conclude that the aquarium population of guppies shows the same mating preference as observed in the natural population.