Final answer:
The Z-value for a two-tail test for proportions is 2.31, which is used to compute the p-value for the hypothesis test. In a standard normal distribution, the area corresponding to the tails beyond Z=2.31 yields the p-value, which is doubled in a two-tailed test.
Step-by-step explanation:
The value of Z for a two-tail test for proportions is given as Z=2.31. In the context of hypothesis testing, this Z value would be used to determine the p-value for the test. Typically, you'd look up the corresponding area under the standard normal curve for a Z-score of 2.31 in a standard normal distribution table or use statistical software to find it. Given that this is a two-tailed test, we would find the probability of obtaining a Z-score that is more extreme than 2.31 in either direction.
To illustrate, if we were to find the area to the right of Z=0.6, we would have P(Z > 0.6) = 0.2742531. Since it's a two-tailed test, we'd double this area to account for both tails, resulting in a p-value of 2 x 0.2742531 = 0.5485062. However, with a Z-score of 2.31, the p-value will be much smaller, signifying a more significant result in a hypothesis test. For example, a Z-score of 3.32, which is also positive, would give us a one-tailed p-value of approximately 0.0103. Doubling this for a two-tailed test would produce a p-value of approximately 0.0206, subject to the exact tail areas for Z=3.32. For Z=2.31, you would expect the p-value to be larger than 0.0206 but still indicate statistical significance if it's below the chosen alpha level, for instance, 0.05.