Final answer:
To calculate the confidence intervals, use the sample mean of 35 and the known population standard deviation of 8. The 90%, 95%, and 99% confidence intervals are calculated using their respective Z-values, resulting in intervals of (33.43, 36.57), (33.13, 36.87), and (32.55, 37.45) respectively.
Step-by-step explanation:
To provide a 90%, 95%, and 99% confidence interval for the population mean based on a sample mean of 35 derived from a simple random sample of 70 items, and assuming the population has a known standard deviation of 8, we can use the following formula for each confidence interval:
Confidence Interval = Sample Mean ± (Z-value * (Population Standard Deviation / √n))
Where Ýan is the square root of the sample size (n=70).
First, we need to find the Z-values corresponding to the required confidence levels. Relevant Z-values for the confidence levels are approximately:
- 1.645 for 90% confidence
- 1.960 for 95% confidence
- 2.576 for 99% confidence
Now, we calculate the error bound (EBM) using the formula EBM = Z-value * (Population Standard Deviation / √n).
For 90% confidence interval:
EBM = 1.645 * (8 / √70)
Substituting the values, we get EBM90 = 1.57
So, the 90% confidence interval is (35 - 1.57, 35 + 1.57) or (33.43, 36.57).
For 95% confidence interval:
EBM = 1.960 * (8 / √70)
Substituting the values, we get EBM95 = 1.87
So, the 95% confidence interval is (35 - 1.87, 35 + 1.87) or (33.13, 36.87).
For 99% confidence interval:
EBM = 2.576 * (8 / √70)
Substituting the values, we get EBM99 = 2.45
So, the 99% confidence interval is (35 - 2.45, 35 + 2.45) or (32.55, 37.45).