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.

Here are a few hints:

1. Use a one-sample z-test for both A and B, which is:
z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)

2. Find the critical or cutoff value to reject the null using a z-table for .05 level of significance for a two-tailed test. If the test statistic exceeds the critical value you find in the table, reject the null and conclude a difference. If the test statistic does not exceed the critical value you find in the table, do not reject the null. You cannot conclude a difference in this case.

I hope this will help get you started.

To determine whether the given samples represent populations that are different from the general population, we can perform a Z-test. A Z-test compares the sample mean to the population mean and determines the likelihood of observing such a sample mean if the populations were actually the same.

Let's go through each sample and perform the Z-test step-by-step:

Sample A:
- Sample size (n_A) = 10
- Sample mean (x̄_A) = 44
- Population mean (μ) = 40
- Population standard deviation (σ) = 6
- Significance level (α) = 0.05 (two-tailed test)

Step 1: State the hypotheses
- Null hypothesis (H0): The sample represents a population that is not different from the general population (μ_A = μ).
- Alternative hypothesis (Ha): The sample represents a population that is different from the general population (μ_A ≠ μ).

Step 2: Set up the rejection region
Since it is a two-tailed test with a significance level of 0.05, we need to split the alpha level in half to get α/2 = 0.025 for each tail. The rejection region will be the extreme 2.5% on each side of the distribution.

Step 3: Calculate the test statistic
The test statistic (Z) can be calculated using the formula:
Z = (x̄_A - μ) / (σ / √n_A)
Z = (44 - 40) / (6 / √10) ≈ 2.31

Step 4: Determine the critical value
To determine the critical value(s), we refer to the Z-table or use a statistical software. For a two-tailed test with α/2 = 0.025, the critical value is approximately ±1.96.

Step 5: Make a decision
Since the calculated test statistic (Z = 2.31) falls in the rejection region (Z > 1.96), we reject the null hypothesis. This means that the sample A represents a population that is different from the general population.

Now let's move on to Sample B:

Sample B:
- Sample size (n_B) = 1
- Sample mean (x̄_B) = 48
- Population mean (μ) = 40
- Population standard deviation (σ) = 6
- Significance level (α) = 0.05 (two-tailed test)

Step 1: State the hypotheses
- Null hypothesis (H0): The sample represents a population that is not different from the general population (μ_B = μ).
- Alternative hypothesis (Ha): The sample represents a population that is different from the general population (μ_B ≠ μ).

Step 2: Set up the rejection region
Since it is a two-tailed test, with α = 0.05, we will use a critical value of ±1.96.

Step 3: Calculate the test statistic
The test statistic (Z) can be calculated using the formula:
Z = (x̄_B - μ) / (σ / √n_B)
Z = (48 - 40) / (6 / √1) = 8 / 6 ≈ 1.33

Step 4: Determine the critical value
For a two-tailed test with α/2 = 0.025, the critical value is approximately ±1.96.

Step 5: Make a decision
Since the calculated test statistic (Z = 1.33) does not fall in the rejection region (|Z| < 1.96), we fail to reject the null hypothesis. This means that the sample B does not represent a population that is different from the general population.

In conclusion, based on the Z-test results, sample A represents a population that is different from the general population, while sample B does not represent a population that is different from the general population.