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 H_{0}: μ = k, we reject H_{0} whenever k falls outside the c = 1 − α confidence interval for μ based on the sample data. When k falls within the c = 1 − α confidence interval, we do not reject H_{0}.
(A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as p, μ_{1} − μ_{2}, or p_{1} − p_{2}, 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 H_{0}. For example, consider a two-tailed hypothesis test with α = 0.01 and
H_{0}: μ = 20 H_{1}: μ ≠ 20
A random sample of size 38 has a sample mean x = 23 from a population with standard deviation σ = 6.
(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 H_{0} 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 H_{0}?
We 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. We fail to 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.
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).These results are the same. We rejected the null hypothesis in part (b) but failed to reject the null hypothesis in part (a).
a)value of c =0.99
sample mean 'x̄= | 23.000 |
sample size n= | 38.00 |
std deviation σ= | 6.000 |
std error ='σx=σ/√n= | 0.9733 |
for 99 % CI value of z= | 2.576 | |
margin of error E=z*std error = | 2.51 | |
lower bound=sample mean-E= | 20.49 | |
Upper bound=sample mean+E= | 25.51 |
k =20
No
We reject the null hypothesis since μ = 20 is not contained in this interval.
b)
p value =0.0020 (please try 0.0021 if this comes wrong)
.We reject the null hypothesis since there is sufficient evidence that μ differs from 20.
we reject in both parts
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