Testing a Claim about Population Proportion

The null hypothesis states the assumed proportion value, typically 0.5, representing no preference or equal likelihood between outcomes. The alternative hypothesis proposes a different proportion, such as a higher preference for one outcome. In the guppy experiment, the null hypothesis assumed equal female preference for orange and blue males, while the alternative hypothesis claimed females would prefer orange males more frequently.

The sample proportion is calculated by dividing the number of successes by the total sample size. In the guppy study, ten females out of twelve preferred orange males, yielding a sample proportion of 0.83. This value is essential for computing the z statistic, which determines whether the observed data significantly differs from the null hypothesis assumption.

The sample must be random and unbiased with binomial data (two possible outcomes). The sample size must be large enough, the probability should be close to 0.5, and the product of sample size and proportion (np) must exceed 5. These requirements allow the binomial distribution to be approximated by the normal distribution for accurate z statistic calculation.

The z statistic is compared to the critical region at a specified significance level, typically 0.05. If the test statistic falls within the critical region, the null hypothesis is rejected. The P-value provides the probability of observing the sample data if the null hypothesis were true. In the guppy experiment, the P-value of 0.011 indicated strong evidence against the null hypothesis.

The normal approximation method requires np greater than 5 and uses z distribution to calculate critical values and P-values. The exact binomial method uses the binomial distribution directly without approximation and does not require the np condition. While both methods yield equally accurate inferences, the exact binomial method is more tedious and typically requires statistical software.

The test statistic fell within the critical region at the 0.05 significance level, and the P-value of 0.011 was less than 0.05. This provided strong statistical evidence to reject the null hypothesis of equal preference. Therefore, the data supported the alternative hypothesis that aquarium females prefer orange males, matching natural population behavior.

Use the exact binomial method when the np product is less than or equal to 5, when the sample size is small, or when high precision is required. This method does not rely on normal distribution assumptions and calculates exact probabilities of obtaining specific success counts. Although more computationally demanding, it provides accurate results without approximation limitations.