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 *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 in later sections.)
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* = 0.05 and

*H*_{0}: *μ* = 21

*H*_{1}: *μ* ≠ 21

A random sample of size 14 has a sample mean *x* = 19
from a population with standard deviation *σ* = 5.

(a) What is the value of *c* = 1 − *α*?

**0.95**

Using the methods of Chapter 7, construct a 1 − α confidence interval for μ from the sample data. (Round your answers to two decimal places.)

Lower Limit: **16.38**

Upper Limit: **21.62**

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

k = **21**

Is this value in the confidence interval?

**Yes**

Do we reject or fail to reject *H*_{0} based on
this information?

**Fail to reject, since μ = 21 is contained in this
interval. **

(b) Using methods of Chapter 8, find the *P*-value for
the hypothesis test. (Round your answer to four decimal
places.)

Answer #1

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...

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Ha: μ ≠ 42. The p-value for this test is
0.03, which is less than 0.05, so the null hypothesis will be
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