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Need help doing this problem, working on this using R programming language
Answer the following questions:

4a.) Suppose we are sampling from a N(µ, 2 =16) distribution. How large must n be so that a 90% CI for µ has length equal to 0.5?

4b.) Suppose you have a random sample from a N(µ,2) distribution with 2 unknown. Let n = 10. Consider testing H0 : µ = 22 versus HA : µ != 22. Suppose you observe x_bar = 20.7 and s2 = 4.17. Consider testing this hypothesis by using confidence intervals. Do you reject H0 at =0.10?, at =0.05?, at =0.01?

4c.) Using the data in part (b) of this problem, perform the T-test in the usual fashion. Use the pt command to find the exact p-value. Is this consistent with your results in part (b)?

Explanation / Answer

(4a)n=693

(1-alpha)*100% confidence interval for population mean=sample mean± z(alpha/2)*sd/sqrt(n)

90% confidence interval for population mean=sample mean±z(0.1/2)*sd/sqrt(n)

here confidence interval length=0.5

margin of error=confidence interval/2=0.5/2=0.25

so margin of error=z(0.1/2)*sd/sqrt(n)=0.25

or,1.6449*4/sqrt(n)=0.25

or, sqrt(n)=1.6449*4/0.25=26.3184

or, n=692.7( 693 next whole number)

(4b)If theconfidence interval does not contain the null hypothesis value, the results are statisticallysignificant.

(1-alpha)*100% confidence interval for population mean=sample mean±t(alpha/2,n-1)*sd/sqrt(n)

90% confidence interval for population mean=mean±t(0.1/2, n-1)*s/sqrt(n)=20.7±t(0.1/2,9)*sqrt(4.17/10)=

20.7± 1.833*sqrt(4.17/10)=20.7±1.2=(19.5,21.9)

95% confidence interval for population mean=mean±t(0.05/2, n-1)*s/sqrt(n)=20.7±t(0.1/2,9)*sqrt(4.17/10)=

20.7± 2.262*sqrt(4.17/10)=20.7±1.5=(19.2,22.2)

99% confidence interval for population mean=mean±t(0.01/2, n-1)*s/sqrt(n)=20.7±t(0.1/2,9)*sqrt(4.17/10)=

20.7±3.25*sqrt(4.17/10)=20.7±2.1=(18.6,22.8)

since only at alpha=0.1 does not have the null hypothesis value so it is significant

(4c) t=|(x_bar - µ)|/(s/sqrt(n))=|(20.7-22)|/(sqrt(4.17/10)=2.01 wtih n-1=10-1=9 df

p-value=0.0749 ( using ms-excel command=TDIST(2.01,9,2))

it is not consistent with the part (b)

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