Question

a genetic experiment involving peas yielded one sample of offspring consisting of 439 green peas and 171 yellow peas. use a 0.01 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.

h₁: p > 0.26
h₁: p < 0.26
e. h₀: p = 0.26
f. h₀: p = 0.26
h₁: p < 0.26
h₁: p > 0.26
what is the test statistic?
z = 1.14
(round to two decimal places as needed.)
what is the p - value?
p - value = □
(round to three decimal places as needed.)

Explanation:

Step1: Determine the type of test

Since the claim is about a proportion and we are using the normal distribution as an approximation to the binomial distribution, and we are testing against a single - value \(p = 0.26\), this is a one - sample proportion \(z\) - test. The formula for the \(z\) - statistic is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(\hat{p}=\frac{x}{n}\), \(x\) is the number of successes (yellow peas), and \(n\) is the sample size. Here, \(x = 171\), \(n=439 + 171=610\), \(p = 0.26\), and \(\hat{p}=\frac{171}{610}\approx0.2803\). But we are given \(z = 1.14\).

Step2: Calculate the P - value

For a one - tailed \(z\) - test (since we assume the claim is \(p = 0.26\) and we test against an alternative. If we assume the alternative is \(H_1:p
eq0.26\) (two - tailed), but if we consider the direction from the \(z\) value. Since \(z = 1.14>0\), if it is a two - tailed test, \(P - value=2\times(1 - P(Z\leq|z|))\). If it is a right - tailed test (\(H_1:p>0.26\)), \(P - value = 1 - P(Z\leq z)\). Using the standard normal distribution table or a calculator, \(P(Z\leq1.14)=0.8729\) (from standard normal table: \(P(Z\leq1.14)\) is found by looking up the row \(1.1\) and column \(0.04\)).

If it is a two - tailed test:
\(P - value=2\times(1 - 0.8729)=2\times0.1271 = 0.254\)
If it is a right - tailed test (\(H_1:p>0.26\)):
\(P - value=1 - 0.8729=0.127\)

Answer:

If it is a two - tailed test: \(P - value = 0.254\)
If it is a right - tailed test (\(H_1:p>0.26\)): \(P - value=0.127\)