Final answer:
The z-score for the one-independent sample z-test is approximately 9.09, indicating a significant difference from the population mean. The 95% confidence interval, assuming a z-score of 1.96, would have upper and lower limits of approximately 43.97 and 40.03, respectively.
Step-by-step explanation:
The question involves computing a one-independent sample z-test and determining the upper and lower confidence limits for a given data set in a research study. Given the sample size (n = 50), the sample mean (M = 42), and the population mean and standard deviation (37.5 ± 3.5), the formula to calculate the z-score is:
z = (M - µ) / (σ / √n)
Substituting the values:
z = (42 - 37.5) / (3.5 / √50)
Calculating the standard error (σ / √n) gives:
Standard Error = 3.5 / √50 = 0.495
Then the z-score is calculated as:
z = (42 - 37.5) / 0.495 ≈ 9.09
To find the confidence interval, you would typically use the z-score corresponding to your desired confidence level. For a 95% confidence level, the z-score is usually approximately 1.96. However, since the z-score calculated here is very high (9.09), it indicates a very low p-value, which would suggest that the sample mean is significantly different from the population mean stated in the null hypothesis.
The confidence interval is calculated using the formula:
Confidence Interval = M ± z*(σ / √n)
Assuming we use the z-score for a 95% confidence level (which is 1.96), the confidence interval would be:
Confidence Interval = 42 ± 1.96 * 0.495
The upper and lower limits of the confidence interval are:
Upper Limit = 42 + (1.96 * 0.495) ≈ 43.97
Lower Limit = 42 - (1.96 * 0.495) ≈ 40.03