Question

Suppose I estimate μ, the mean of a population. I obtain estimate 6.5 and 95% confidence interval (4.5, 8.5).

A) Can I reject H0 : μ = 8 using at two-sided test at the 1% significance level?

a) Yes. b) No. c) Not enough information to decide.

B) Can I reject H0 : μ = 8 using at two-sided test at the 10% significance level?

a) Yes. b) No. c) Not enough information to decide.

C) Can I reject H0 : μ = 9 using at two-sided test at the 1% significance level?

a) Yes. b) No. c) Not enough information to decide.

D) Can I reject H0 : μ = 9 using at two-sided test at the 10% significance level?

a) Yes. b) No. c) Not enough information to decide.

Answer 1

Suppose I estimate μ, the mean of a population. I obtain an estimate of 6.5 and a 95% confidence interval (4.5, 8.5).

Here we calculated a 95 % confidence interval.

If the null hypothesis value is included in the confidence interval then we do not reject the null hypothesis at a 5% level of significance.

at a 1% level of significance ( 99% confidence interval will be wider than the 95 % confidence interval), it also includes the null hypothesis value so we can say that do not reject the null hypothesis.

at a 10% level of significance ( 90% confidence interval will be narrower than the 95 % confidence interval), it may or may not includes the null hypothesis value so we cannot make any conclusion from the 90% confidence interval.

But

If the null hypothesis value is not included in the confidence interval then we reject the null hypothesis at a 5% level of significance.

at a 1% level of significance ( 99% confidence interval will be wider than the 95 % confidence interval), it may include the null hypothesis value or may not include the null hypothesis so we cannot make any conclusion at 1% l.o.s.

at a 10% level of significance ( 90% confidence interval will be narrower than the 95 % confidence interval), it also does not includes the null hypothesis value so we can reject the null hypothesis at 90% confidence interval.

Result Table

Confidence interval	a	Include the null hypothesis value	Does not include null hypothesis value
95	0.05	do not reject Ho	Reject Ho
90	0.10	No conclusion	Reject Ho
99	0.01	do not reject Ho	No conclusion

95 % Confidence interval is ( 4.5, 8.5 )

A) Can I reject H0: μ = 8 using at two-sided test at the 1% significance level?

u = 8 is in the confidence interval.so we do not reject Ho at a 5% level of significance.

So also at a 1% level of significance, we do not reject Ho.

Answer:- No

B) Can I reject H0 : μ = 8 using at two-sided test at the 10% significance level?

u = 8 is in the confidence interval.so we do not reject Ho at a 5% level of significance.

So also at a 10% level of significance, we cannot conclude anything.

c) Not enough information to decide.

C) Can I reject H0 : μ = 9 using at two-sided test at the 1% significance level?

u = 9 is not in the confidence interval.so we reject Ho at a 5% level of significance.

So also at a 1% level of significance, we cannot make any inference.

c) Not enough information to decide.

D) Can I reject H0 : μ = 9 using at two-sided test at the 10% significance level?

u = 9 is not in the confidence interval.so we reject Ho at a 5% level of significance.

So also at a 10% level of significance, we reject Ho.

a) Yes.