Agenetic experiment involving peas yielded one sample of offspring consisting of 409 green peas and 171 yellow peas. use a 0.05 significance level to test the claim that under the same circumstances, 26% of offspring peas will be yellow. identify the null hypothesis, alternative hypothesis, test statistic, p-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. use the p-value method and the normal distribution as an approximation to the binomial distribution.

The usual expectation for this kind of experiment is that the peas would yield green and yellow peas in a 3:1 ratio, or 75% green to 25% yellow. So your null hypothesis is that the proportion of yellow peas is .

You're testing the claim that 26% of the offspring will be yellow, which means the alternative hypothesis is that the proportion of yellow peas is actually 0.26, or more generally that the expected proportion is greater than 0.25, or .

The test statistic in this case will be



where is the proportion assumed under the null hypothesis, is the measured proportion, and is the sample size. You have , , and , so the test statistic is



Because you're testing , this is a right-tailed test, so the P-value is



The critical value for a right-tail test at a 0.05 significance level is , which means the rejection region is any test statistic that is larger than this critical value. Since in this case, we reject the null hypothesis.

So the conclusion for this test is that the sample proportion is indeed statistically significantly different from the proportion suggested by the null hypothesis.