Final answer:
a) The 90% confidence interval estimate for the population mean is (2.680, 3.986).
b) Increasing the confidence level to 95% widens the confidence interval, capturing a larger range of possible population means.
Step-by-step explanation:
a) To compute a 90% confidence interval estimate for the population mean, we can use the formula:
CI = sample mean ± (critical value) x (standard error)
First, calculate the sample mean:
Sample mean = (2+5+8+3+1+2+4+3+2)/9
= 3.333
Next, calculate the standard deviation of the sample:
Standard deviation = sqrt((sum of (xi - sample mean) squared)/n-1)
Standard deviation = sqrt((9.333)/8)
≈ 1.15
Then, calculate the standard error:
Standard error = standard deviation/sqrt(n)
Standard error = 1.15/sqrt(9)
= 0.383
Next, find the critical value for a 90% confidence interval using a standard normal distribution table or calculator.
For a 90% confidence interval, the critical value is approximately 1.645.
Lastly, calculate the confidence interval:
CI = 3.333 ± (1.645)(0.383)
= (2.680, 3.986)
Therefore, the 90% confidence interval estimate for the population mean is (2.680, 3.986).
If the confidence level is increased to 95%, we need to find the new critical value.
For a 95% confidence interval, the critical value is approximately 1.96. Using this new critical value:
CI = 3.333 ± (1.96)(0.383) = (2.586, 4.080)
b) The impact of increasing the confidence level to 95% is that the confidence interval becomes wider, meaning it captures a larger range of possible population means.
This occurs because a higher confidence level requires more certainty and therefore a larger range of values to be included in the interval.
Increasing the confidence level to 95% ensures a higher level of confidence that the interval contains the true value of the population mean, but at the cost of a wider interval.