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 mean of 40, a standard 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 significance level, and make a drawing of the distributions involved.
To determine if the given samples are different from the general population, we can carry out a Z-test for each sample. The Z-test will allow us to calculate the Z-score, which will tell us how many standard deviations away from the mean the sample mean is.
Here are the steps to carry out the Z-test:
Step 1: State the hypothesis
- Null Hypothesis (H0): The sample mean is not significantly different from the population mean.
- Alternative Hypothesis (Ha): The sample mean is significantly different from the population mean.
Step 2: Set the significance level
The significance level (alpha) is given as 0.05.
Step 3: Calculate the test statistic
The test statistic for a Z-test is the Z-score. The formula to calculate the Z-score is:
Z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
For sample (a):
Z = (44 - 40) / (6 / sqrt(10))
For sample (b):
Z = (48 - 40) / (6 / sqrt(1))
Step 4: Compare the test statistic with critical values
We need to compare the calculated Z-score with the critical values from the Z-table. Since we are conducting a two-tailed test, we need to find the critical values that correspond to a significance level of 0.025 on each tail.
Step 5: Make a decision
If the calculated Z-score is greater than or less than the critical values obtained in Step 4, we can reject the null hypothesis. If the calculated Z-score falls within the critical values, we fail to reject the null hypothesis.
Regarding the drawing of the distributions involved, we can create a graph showing the normal distribution for the general population and indicate the sample mean and Z-score on the graph.
Please note that I am unable to generate a visual graph in this text-based format, but you can easily create one using statistical software like Excel or online graphing tools.
Repeat the above steps for both sample (a) and (b) to determine if they are significantly different from the general population.