Question

A random sample is selected from a normal population with a mean of μ = 30 and a standard deviation of σ = 8. After a treatment is administered to the individuals in the sample, the sample mean is found to be m = 33. If the sample consists of n = 16 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05. If the sample consists of n = 64 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05. Comparing your answers for parts a and b, explain how the size of the sample influences the outcome of a hypothesis test?

Answer 1

The sample mean is sufficient to conclude that the treatment has a significant effect when the sample size is 64, but not when the sample size is 16. The size of the sample influences the outcome of a hypothesis test.

Step-by-step explanation:

To determine if the treatment has a significant effect, we can perform hypothesis tests using the sample mean. Since the population standard deviation is known (σ = 8) and the sample size is less than 30, we can use a z-test. For a two-tailed test with α = .05, the critical z-value is ±1.96.

For the sample with n = 16, the test statistic is (33 - 30) / (8 / √16) = 1.5. Since 1.5 < 1.96, we fail to reject the null hypothesis and conclude that the treatment does not have a significant effect.

For the sample with n = 64, the test statistic is (33 - 30) / (8 / √64) = 3. Since 3 > 1.96, we reject the null hypothesis and conclude that the treatment has a significant effect.

The size of the sample influences the outcome of a hypothesis test because a larger sample size reduces the variability of the sample mean and increases the power of the test. With a larger sample size, even smaller differences between the sample mean and the population mean can be detected, leading to a higher likelihood of rejecting the null hypothesis.