Question

QUESTIONS 3 AND 4 ARE BASED ON THE FOLLOWING INFORMATION.

The number of times the AI algorithm is successful at detecting fake news is normally distributed
with a sample mean of 900 and the sample standard deviation of 30. Assume a sample size of
100 was used.

Question 3

What is the 95% confidence interval (CI) estimate for the population mean?

(1) (862.6636; 937.3364)
(2) (864.2155; 935.7845)
(3) (852.8959; 947.1041)
(4) (869.9665; 930.0335)
(5) None of the above.

Question 4

What is the 99% confidence interval estimate for the population mean?

(1) (862.6636; 937.3364)
(2) (864.2155; 935.7845)
(3) (852.8959; 947.1041)
(4) (869.9665; 930.0335)
(5) None of the above

Answer 1

$Question\;3\\ Given \; that, \bar x=900, s=30, n=100, \\ \text{The confidence interval for the mean }\\ \text{can be calculated as follows:}\\ \bar x±Z_{\frac{α}{2}}(\frac{s}{\sqrt{n}}),\\ Z_{\frac{α}{2}}=Z_{\frac{0.05}{2}}=Z_{0.025}=1.96,\\ \text{ the lower limit }= 900-1.96(\frac{30}{\sqrt{100}})\\ =894.12,\\ \text{ the upper limit} = 900+1.96(\frac{30}{\sqrt{100}})\\ =905.88,\\ \text{so the confidence interval is}\\ 894.12\leq \bar x \leq 905.88\\ \text{the answer is (5) None of the above}\\ Question\;4\\ Z_{\frac{α}{2}}=Z_{\frac{0.01}{2}}=Z_{0.005}=2.58,\\ \text{ the lower limit }= 900-2.58(\frac{30}{\sqrt{100}})\\ =892.26,\\ \text{ the upper limit} = 900+2.58(\frac{30}{\sqrt{100}})\\ =907.74,\\ \text{so the confidence interval is}\\ 892.26\leq \bar x \leq 907.74\\ \text{the answer is (5) None of the above}\\$