A random sample of 10 independent observations from a normally distributed population yielded the following values 51, 53, 49, 43, 47, 46, 45, 30, 60, 52.
a) Estimate the standard deviation of the sampling distribution of means for samples like this one.
b) Using á = .05, test the hypothesis that the true mean is 50 against the alternative that the true mean is not 50.
c) State the conclusion.
a) Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
Standard deviation of the sampling distribution of means = SEm = SD/√n
b) Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
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 proportion/probability related to the Z score.
c) Depends on your calculations.
a) To estimate the standard deviation of the sampling distribution of means, we can use the formula:
Standard deviation of the sampling distribution of means = standard deviation of the population / square root of the sample size.
We know that the sample size is 10 (n = 10). To find the standard deviation of the population, we need to find the standard deviation of the sample. Here's how you can do it:
Step 1: Find the mean of the sample. Sum up all the values and divide by the sample size:
Mean = (51 + 53 + 49 + 43 + 47 + 46 + 45 + 30 + 60 + 52) / 10 = 48.6
Step 2: Find the squared difference between each value and the mean. Square each difference:
(51 - 48.6)^2, (53 - 48.6)^2, (49 - 48.6)^2, (43 - 48.6)^2, (47 - 48.6)^2, (46 - 48.6)^2, (45 - 48.6)^2, (30 - 48.6)^2, (60 - 48.6)^2, (52 - 48.6)^2
Step 3: Sum up all the squared differences:
Sum = (51 - 48.6)^2 + (53 - 48.6)^2 + (49 - 48.6)^2 + (43 - 48.6)^2 + (47 - 48.6)^2 + (46 - 48.6)^2 + (45 - 48.6)^2 + (30 - 48.6)^2 + (60 - 48.6)^2 + (52 - 48.6)^2
Step 4: Divide the sum by (n - 1) to find the sample variance:
Variance = Sum / (n - 1)
Step 5: Take the square root of the variance to find the sample standard deviation:
Standard deviation of the sample = √(Variance)
Using these steps, calculate the sample standard deviation.
b) To test the hypothesis that the true mean is 50 against the alternative that the true mean is not 50, we can use a t-test. Here's how you can do it:
Step 1: Set up the null and alternative hypotheses:
Null hypothesis (H0): The true mean is 50.
Alternative hypothesis (Ha): The true mean is not 50.
Step 2: Determine the significance level (α). In this case, it is given that α = 0.05.
Step 3: Calculate the t-statistic using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / √(sample size))
The sample mean is the mean calculated in part a). The hypothesized mean is 50. The sample standard deviation is the value obtained in part a), and the sample size is 10.
Using these values, calculate the t-statistic.
Step 4: Find the degrees of freedom. For a sample size of n = 10, the degrees of freedom are (n - 1) = 9.
Step 5: Determine the critical value corresponding to the chosen significance level and degrees of freedom. You can refer to a t-distribution table or use statistical software for this.
Step 6: Compare the calculated t-statistic with the critical value. If the calculated t-statistic falls in the rejection region (outside the critical region), reject the null hypothesis; otherwise, fail to reject the null hypothesis.
c) State the conclusion based on the result obtained in step 6. If the null hypothesis is rejected, you can conclude that there is evidence to suggest that the true mean is not 50. If the null hypothesis is not rejected, you can conclude that there is not enough evidence to suggest that the true mean is different from 50.