Answer to Question #105326 in Statistics and Probability for Katenda

Question

@ refers to confidence interval estimate for the population mean

X with a mean 100 and a population standard deviation of 9. A sample of size 30 was used. Answer 1 to 4

Q1
(A) 90% @ is (97.2970; 102.7030)
(B) 95% @ is (96.7794; 103.2206)
(C) 99% @ is (95.7606; 104.2394)

Which is correct?
(1) A
(2) B
(3) C
(4) A and B
(5) A, B and C

Q2
If sample size stays the same, what happens to confidence interval estimate if level of confidence increases
(1) don’t change
(2) become wider
(3) become narrower
(4) converge to 0
(5) None

Q3
Which is incorrect
(1) If sample size(SS) is 60, 95% @ is (97.7227; 102.2773)
(2) If SS is 100, 95% @ is (98.2360; 101.7640)
(3) If SS is 200, 95% @ is (98.7527; 101.2473)
(4) If SS is 500, 95% @ is (99.2111; 100.7889)
(5) None

Q4
If level of significance stays the same, what happens to confidence interval estimate if sample size increases
(1) don’t change
(2) become wider
(3) become narrower
(4) converge to 0
(5) None

Answer 1

Q1

$$CI=(\bar{X}-z_{\alpha/2}{\sigma \over \sqrt{n}},\ \bar{X}+z_{\alpha/2}{\sigma \over \sqrt{n}})$$

Given that $\bar{X}=100, \sigma=9, n=30$

$$90\%:z_{\alpha/2}=1.645$$ $$CI=(100-1.645{9 \over \sqrt{30}},\ 100+1.645{9 \over \sqrt{30}})$$ $$CI=(97.2970,\ 102.7030)$$

$$95\%:z_{\alpha/2}=1.96$$ $$CI=(100-1.96{9 \over \sqrt{30}},\ 100+1.96{9 \over \sqrt{30}})$$ $$CI=(96.7794,\ 103.2206)$$

$$99\%:z_{\alpha/2}=2.576$$ $$CI=(100-2.576{9 \over \sqrt{30}},\ 100+2.576{9 \over \sqrt{30}})$$ $$CI=(95.7672,\ 104.2328)$$

Correct:

(4) A and B 

Q2

If sample size stays the same, what happens to confidence interval estimate if level of confidence increases 

Correct:

(2) become wider 

Q3 

$$95\%:z_{\alpha/2}=1.96$$

$$n=60:\ CI=(100-1.96{9 \over \sqrt{60}},\ 100+1.96{9 \over \sqrt{60}})$$ $$CI=(97.7227,\ 102.2773)$$

$$n=100:\ CI=(100-1.96{9 \over \sqrt{100}},\ 100+1.96{9 \over \sqrt{100}})$$ $$CI=(98.2360,\ 101.7640)$$

$$n=200:\ CI=(100-1.96{9 \over \sqrt{200}},\ 100+1.96{9 \over \sqrt{200}})$$ $$CI=(98.7527,\ 101.2473)$$

$$n=500:\ CI=(100-1.96{9 \over \sqrt{500}},\ 100+1.96{9 \over \sqrt{500}})$$ $$CI=(99.2111,\ 100.7889)$$

Which is incorrect 

(5) None 

Q4

If level of significance stays the same, what happens to confidence interval estimate if sample size increases 

(3) become narrower