Final answer:
When performing a hypothesis test for the mean using a t-test instead of a z-test, the p-value would be larger due to the heavier tails of the t-distribution. This indicates less evidence against the null hypothesis in the t-test compared to the z-test, which is used for larger samples or when the population standard deviation is known.
option b is the correct
Step-by-step explanation:
If you perform a hypothesis test for the mean and use a t-test instead of a z-test, and everything else remains the same, your p-value for the t-test would generally be larger compared to the p-value for the z-test. The reason for this difference stems from the distributions that the two tests use.
The t-distribution is similar to the normal distribution but has heavier tails, which means there is a greater probability of observing values far away from the mean. Because of these heavier tails, when using the same test statistic, a t-distribution will yield a larger p-value compared to a normal distribution.
For example, if the calculated test statistic for the p-value is -2.08 and assuming we are dealing with the same degrees of freedom for the t-test, the area in the tails (or the p-value) is larger in a t-distribution compared to a normal distribution. This higher p-value suggests less evidence against the null hypothesis when using a t-test relative to a z-test. The choice between using a z-test and a t-test usually depends on the sample size and whether the population standard deviation is known. Z-tests are often used when the sample size is large (typically over 30) and the population standard deviation is known, while t-tests are used for smaller sample sizes or when the population standard deviation is unknown.