For each of the following samples that were given an experimental treatment, test whether these samples represent populations that are different from the general population: (a) a sample of 10 with a mean of 44, (b) a sample of 1 with a mean of 48. The general population of individuals has a sample of 40, a standad deviation of 6, and follows a normal curve. For each sample, carry out a Z test using the five steps hypothesis testing with a two tailed test at the .05 signifigance level
To test whether the given samples represent populations that are different from the general population, we can carry out a Z-test using the five-step hypothesis testing process. Here are the steps to perform this test in both cases:
(a) Sample of 10 with a mean of 44:
Step 1: Formulate the null and alternative hypotheses.
Null hypothesis (H0): The sample mean is not significantly different from the mean of the general population.
Alternative hypothesis (Ha): The sample mean is significantly different from the mean of the general population.
Step 2: Set the significance level.
The given significance level is 0.05 (or 5%).
Step 3: Calculate the test statistic.
The test statistic, Z, can be calculated using the formula:
Z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
Here, the sample mean is 44, the population mean is the mean of the general population (unknown from the given information), the population standard deviation is 6, and the sample size is 10.
Z = (44 - population mean) / (6 / sqrt(10))
Step 4: Determine the critical value(s).
Since it's a two-tailed test, we need to find the critical value(s) for rejection region(s) on both sides of the distribution.
At a significance level of 0.05, the critical value for a two-tailed test would be ±1.96.
Step 5: Make a decision and interpret the result.
If the calculated Z-value falls within the rejection region (less than -1.96 or greater than +1.96), we reject the null hypothesis and conclude that the sample mean is significantly different from the mean of the general population.
(b) Sample of 1 with a mean of 48:
The process for this sample would be similar to the previous case:
Step 1: Formulate the null and alternative hypotheses.
H0: The sample mean is not significantly different from the mean of the general population.
Ha: The sample mean is significantly different from the mean of the general population.
Step 2: Set the significance level.
The given significance level is 0.05 (or 5%).
Step 3: Calculate the test statistic.
Using the same formula as above, the test statistic, Z, can be calculated. Here, the sample mean is 48, the population mean is unknown, the population standard deviation is 6, and the sample size is 1.
Z = (48 - population mean) / (6 / sqrt(1))
Step 4: Determine the critical value(s).
For a two-tailed test at a significance level of 0.05, the critical value would still be ±1.96.
Step 5: Make a decision and interpret the result.
If the calculated Z-value falls within the rejection region (less than -1.96 or greater than +1.96), we reject the null hypothesis and conclude that the sample mean is significantly different from the mean of the general population.
Note:
While the Z-test can be applied in both cases, it is important to note that the power of the test (ability to detect a difference) will be higher with a larger sample size. In the case of a sample size of 1 (b), it may be more appropriate to use alternative statistical tests such as a t-test or non-parametric tests, depending on the specific context.