Below are random sample of 4 observations.
25,75,32,56
Based on the given information, determine whether or not the population from which this sample was taken is significantly different from 48 at 95 percent confidence level. (Assume the population is normally distributed).
To determine whether the population from which this sample was taken is significantly different from 48 at a 95 percent confidence level, we can conduct a hypothesis test.
Step 1: State the null and alternative hypotheses:
Null Hypothesis (H0): The population mean is equal to 48.
Alternative Hypothesis (Ha): The population mean is not equal to 48.
Step 2: Calculate the sample mean and sample standard deviation:
For the given sample observations: 25, 75, 32, 56
Sample mean (x̄) = (25 + 75 + 32 + 56) / 4 = 47
Sample standard deviation (s) = √[((25-47)² + (75-47)² + (32-47)² + (56-47)²) / 3] = √[(5068 / 3)] ≈ 21.53
Step 3: Perform the hypothesis test using a t-test:
We need to calculate the t-value, which measures the difference between the sample mean and the hypothesized population mean relative to the variability in the sample.
t-value = (x̄ - μ) / (s / √n)
where:
x̄ = sample mean
μ = hypothesized population mean
s = sample standard deviation
n = sample size
In this case, x̄ = 47, μ = 48, s ≈ 21.53, and n = 4.
t-value = (47 - 48) / (21.53 / √4) = -1 / (21.53 / 2) ≈ -0.093
Step 4: Determine the critical t-value:
Since the desired confidence level is 95 percent, we need to find the critical t-value at alpha = 0.05 significance level, with degrees of freedom (df) = n - 1.
For this sample size (n = 4), the critical t-value can be obtained from a t-distribution table or a statistical software. For a two-tailed test at alpha = 0.05 with df = 3, the critical t-value is approximately 3.18.
Step 5: Compare the t-value with the critical t-value:
If the absolute value of the calculated t-value is greater than the critical t-value, we reject the null hypothesis; otherwise, we fail to reject it.
In this case, the absolute value of the calculated t-value is 0.093, which is less than the critical t-value of 3.18.
Step 6: Conclusion:
Since the absolute value of the calculated t-value is not greater than the critical t-value, we fail to reject the null hypothesis. Therefore, we do not have enough evidence to suggest that the population mean is significantly different from 48 at a 95 percent confidence level.