Question

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let c be the level of confidence used to construct a confidence interval from sample data. Let α be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean.

For a two-tailed hypothesis test with level of significance α and null hypothesis H₀: μ = k, we reject H₀ whenever k falls outside the c = 1 − α confidence interval for μbased on the sample data. When k falls within the c = 1 − α confidence interval, we do not reject H₀.

(A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as p, μ₁ − μ₂, or p₁ − p₂, which we will study later.) Whenever the value of k given in the null hypothesis falls outside the c = 1 − α confidence interval for the parameter, we reject H₀. For example, consider a two-tailed hypothesis test with α = 0.01 and

H₀: μ = 20        H₁: μ ≠ 20

A random sample of size 32 has a sample mean x = 22 from a population with standard deviation σ = 4.

(a) What is the value of c = 1 − α?


Construct a 1 − α confidence interval for μ from the sample data. (Round your answers to two decimal places.)

lower limit
upper limit

What is the value of μ given in the null hypothesis (i.e., what is k)?
k =  

Is this value in the confidence interval?

YesNo     

Do we reject or fail to reject H₀ based on this information?

We fail to reject the null hypothesis since μ = 20 is not contained in this interval.We fail to reject the null hypothesis since μ = 20 is contained in this interval.     We reject the null hypothesis since μ = 20 is not contained in this interval.We reject the null hypothesis since μ = 20 is contained in this interval.

(b) Using methods of this chapter, find the P-value for the hypothesis test. (Round your answer to four decimal places.)


Do we reject or fail to reject H₀?

We reject the null hypothesis since there is insufficient evidence that μ differs from 20.We fail to reject the null hypothesis since there is sufficient evidence that μ differs from 20.     We fail to reject the null hypothesis since there is insufficient evidence that μ differs from 20.We reject the null hypothesis since there is sufficient evidence that μ differs from 20.

Compare your result to that of part (a).

We rejected the null hypothesis in part (a) but failed to reject the null hypothesis in part (b).We rejected the null hypothesis in part (b) but failed to reject the null hypothesis in part (a).     These results are the same.

Answer 1

given data are:-

sample mean () = 22

population sd () = 4

sample size (n) = 32

level of significance ()= 0.01

hypothesis:-

c = (1- ) = (1 - 0.01) = 0.99

the 99 % confidence interval for be:-

[ we know that, for 99% confidence level, both tailed test, z = 2.576 ]

a 1 − α confidence interval for μ from the sample data be:-

lower limit	20.18
upper limit	23.82

the value of μ given in the null hypothesis be:-

k = 20

NO, this value is not in the confidence interval

based on this information :-

We reject the null hypothesis since μ = 20 is not contained in this interval

b). the test statistic be:-

p value = 0.0047

[ using ti 84 plus calculator.

steps:-

2ND vars select normalcdf in lower type 2.83 , in upper type 1000, in type 0 ,in type 1 enterenter.

you will get 0.002327

this is one tailed p value.

since, this is a both tailed test , the p value be:-

(0.002327*2) = 0.004654 0.0047 ]

p value = 0.0047 < 0.01.

so ,our decision be:-

We reject the null hypothesis since there is sufficient evidence that μ differs from 20.

c). comparison of results obtained from a and b be:-

These results are the same.

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