This is the first question..... For a population that has a standard deviation of 10, figure the standard deviation of the distribution of means for samples of size- (b) 3.
This is the second question....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 answer the first question, you want to calculate the standard deviation of the distribution of means for samples of size 3 from a population with a standard deviation of 10.
To calculate the standard deviation of the distribution of means (also known as the standard error), you can use the formula:
Standard Error = Population Standard Deviation / √(Sample Size)
In this case, the population standard deviation is given as 10, and the sample size is 3. Plugging these values into the formula:
Standard Error = 10 / √(3)
Calculating the square root of 3, the standard error is approximately 5.77. Therefore, the standard deviation of the distribution of means for samples of size 3 is approximately 5.77.
To answer the second question, you are asked to test whether two different samples represent populations that are different from the general population.
To carry out the Z test, which compares sample means to a known population mean, you need to follow these steps:
Step 1: State the Hypotheses
- Null Hypothesis (H0): The sample mean is not significantly different from the population mean.
- Alternative Hypothesis (H1): The sample mean is significantly different from the population mean.
Step 2: Set the Significance Level
- In this case, the significance level (α) is given as 0.05.
Step 3: Calculate the Test Statistic
- The test statistic for comparing means is the Z-score. The formula for a Z-score is:
Z = (Sample Mean - Population Mean) / (Population Standard Deviation / √(Sample Size))
Step 4: Determine the Critical Value
- The critical value is determined based on the significance level, the sample size, and the type of test (two-tailed in this case).
Step 5: Compare the Test Statistic with the Critical Value
- If the test statistic falls within the rejection region (outside the critical value range), you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.
For each sample, follow these steps and compare the calculated Z-score with the critical value.
For sample (a) with a sample size of 10 and a mean of 44:
- Z = (44 - 40) / (6 / √(10))
- Calculate the Z-score and compare it with the critical value at a 0.05 significance level.
For sample (b) with a sample size of 1 and a mean of 48:
- Z = (48 - 40) / (6 / √(1))
- Calculate the Z-score and compare it with the critical value at a 0.05 significance level.
Additionally, to make a drawing of the distributions involved, you can plot the normal distribution curve for the general population as well as the distribution of each sample.