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 a t the .05 significance level, and make a drawing of the distributions involved.

To determine whether the given sample means differ significantly from the general population mean, we will conduct a Z-test for each sample. The general steps for hypothesis testing with a Z-test are as follows:

Step 1: State the hypotheses
- 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 (α)
In this case, the significance level is stated as α = 0.05.

Step 3: Compute the test statistic
The formula for the Z-test statistic is:
Z = (sample mean - population mean) / (population standard deviation / square root of sample size)

For sample (a):
Z = (44 - 40) / (6 / √10) = 1.943

For sample (b):
Z = (48 - 40) / (6 / √1) = 2.667

Step 4: Determine the critical value(s)
Since this is a two-tailed test, we need to find the critical values associated with a significance level of 0.025 for each tail.
Using a Z-table or calculator, we find that the critical value for a two-tailed test with α = 0.025 is approximately ±1.96.

Step 5: Make a decision and draw conclusions
- For sample (a): The test statistic (Z = 1.943) does not exceed the critical value of ±1.96. Therefore, we fail to reject the null hypothesis. It means that the sample with a mean of 44 does not represent a population significantly different from the general population.
- For sample (b): The test statistic (Z = 2.667) exceeds the critical value of ±1.96. Therefore, we reject the null hypothesis. It means that the sample with a mean of 48 represents a population significantly different from the general population.

Drawing of the distributions involved:
For both samples (a) and (b), the distributions will be normal curves. However, since we are only comparing the means, it is sufficient to represent the distributions with the mean and standard deviation.

Sample (a):
Sample Mean = 44, Population Mean = 40, Population Standard Deviation = 6

44
|
|
|
|_____________________
40 (General Population Mean)

Sample (b):
Sample Mean = 48, Population Mean = 40, Population Standard Deviation = 6

48
|
|
|
|_____________________
40 (General Population Mean)

In summary, sample (a) with a mean of 44 does not represent a population significantly different from the general population. However, sample (b) with a mean of 48 represents a population significantly different from the general population.

To test whether these samples represent populations that are different from the general population, we can perform a Z-test using the five-step hypothesis testing approach. Here's how you can carry out the Z-test for each sample:

Step 1: State the hypotheses
- We assume that the general population has a mean of 40, and we want to test if the sample means are significantly different from this population mean.
- The null hypothesis (H₀) states that there is no significant difference between the sample mean and the population mean.
- The alternative hypothesis (H₁) states that there is a significant difference between the sample mean and the population mean.

For sample a:
H₀: The sample mean of 44 is not significantly different from the population mean of 40.
H₁: The sample mean of 44 is significantly different from the population mean of 40.

For sample b:
H₀: The sample mean of 48 is not significantly different from the population mean of 40.
H₁: The sample mean of 48 is significantly different from the population mean of 40.

Step 2: Set the significance level
- The significance level (α) is given as .05, which means we will reject the null hypothesis if the probability of observing the sample mean is less than 5% assuming the null hypothesis is true.

Step 3: Collect and summarize the data
- The sample size for sample a is 10, and the sample mean is 44.
- The sample size for sample b is 1, and the sample mean is 48.
- The population mean is 40, and the standard deviation is 6.

Step 4: Calculate the test statistic
- For each sample, we will calculate the Z-score, which measures the number of standard deviations the sample mean is away from the population mean.
- The Z-score can be calculated using the formula:
Z = (sample mean - population mean) / (standard deviation / √sample size)

For sample a:
Z = (44 - 40) / (6 / √10)

For sample b:
Z = (48 - 40) / (6 / √1)

Step 5: Make a decision and draw the distributions involved
- We will compare the absolute value of the Z-score to the critical Z-value for a two-tailed test at the 0.05 significance level (α = 0.05).
- If the absolute value of the Z-score is greater than the critical Z-value, we will reject the null hypothesis.

To make a drawing of the distributions involved, we would need to plot normal distributions for both the general population mean of 40 and the sample means of 44 and 48. However, as a text-based bot, I cannot create visual drawings. You can refer to a statistical software or tool that allows you to plot such distributions.

By calculating the Z-scores and comparing them to the critical Z-value, you can determine whether the samples represent populations that are different from the general population.