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.
a. Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores. P = .05 means .025 in each direction.
b. Z = (score-mean)/SD
Use same table.
To determine whether the given samples represent populations that are different from the general population, we can perform a Z test for each sample. Here are the steps for conducting the hypothesis testing:
Step 1: State the null and alternative hypothesis:
- Null Hypothesis (H0): The sample mean is not significantly different from the population mean (μ = 40).
- Alternative Hypothesis (Ha): The sample mean is significantly different from the population mean (μ ≠ 40).
Step 2: Set the significance level:
The significance level is given as α = .05.
Step 3: Calculate the test statistic (Z-score):
The formula to calculate the Z-score is:
Z = (x̄ - μ) / (σ / √n)
where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
For sample (a):
x̄ = 44
μ = 40
σ = 6
n = 10
Using the formula, we can calculate the Z-score for sample (a):
Z = (44 - 40) / (6 / √10)
For sample (b):
x̄ = 48
μ = 40
σ = 6
n = 1
Using the formula, we can calculate the Z-score for sample (b):
Z = (48 - 40) / (6 / √1)
Step 4: Determine the critical value(s):
Since this is a two-tailed test, we need to find the critical values at the chosen significance level (.05/2 = .025).
You can use a standard normal distribution table or a statistical calculator to find the critical value(s) corresponding to the significance level.
Step 5: Compare the test statistic with the critical value(s) and make a decision:
If the absolute value of the calculated Z-score is greater than the critical value(s), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Now, let's calculate the Z-scores and make the distributions' drawings for each sample:
For sample (a):
Z = (44 - 40) / (6 / √10)
Z = 4 / (6 / √10)
Z = 4 / (6 / 3.16)
Z ≈ 4 / 1.897
Z ≈ 2.11
For sample (b):
Z = (48 - 40) / (6 / √1)
Z = 8 / 6
Z = 1.33
The critical value(s) at a significance level of .025 can be found using a standard normal distribution table or a statistical calculator. Let's assume the critical value is Z = ±1.96.
For sample (a):
Z = 2.11 > 1.96 (critical value)
Since the test statistic (Z-score) is greater than the critical value, we reject the null hypothesis. Therefore, sample (a) represents a population that is significantly different from the general population.
For sample (b):
Z = 1.33 < 1.96 (critical value)
Since the test statistic (Z-score) is less than the critical value, we fail to reject the null hypothesis. Therefore, sample (b) does not represent a population that is significantly different from the general population.
Drawing of Distributions:
Sample (a):
The distribution of sample (a) would have a mean of 44, with the general population mean (40) represented by the center of the distribution. The distribution would be a normal curve.
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40
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Sample (b):
The distribution of sample (b) would have a mean of 48, with the general population mean (40) represented by the center of the distribution. The distribution would also be a normal curve.
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40
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Please note that the critical value of ±1.96 and the drawing of distributions are hypothetical examples. The actual critical values may vary depending on the chosen significance level and the statistical calculator or table used.
To test whether the samples represent populations that are different from the general population, you can use the Z test.
Here are the steps you can follow for each sample:
Step 1: State the null and alternative hypotheses
- The null hypothesis (H0) states that there is no difference between the sample mean and the population mean.
- The alternative hypothesis (Ha) states that there is a difference between the sample mean and the population mean.
For sample (a):
H0: µ = 40
Ha: µ ≠ 40
For sample (b):
H0: µ = 40
Ha: µ ≠ 40
Step 2: Define the significance level
The significance level (α) is set at 0.05, which means we're looking for a p-value less than 0.05 to reject the null hypothesis.
Step 3: Calculate the test statistic
To calculate the test statistic (Z-score), you need to use the formula:
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: Find the p-value
To find the p-value, you can use a Z-table or a statistical calculator. The p-value gives the probability of observing the test statistic (or a more extreme value) under the null hypothesis.
Step 5: Make a conclusion
Compare the p-value to the significance level (α). If the p-value is less than α, we reject the null hypothesis and conclude that there is a significant difference between the sample mean and the population mean. If the p-value is greater than α, we fail to reject the null hypothesis.
For each sample, you would follow these steps and then draw the distributions involved, which would be a normal curve for both the sample and population mean.