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 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 H₀. For example, consider a two-tailed hypothesis test with α = 0.05and

H₀: μ = 21
H₁: μ ≠ 21

A random sample of size 33 has a sample mean x = 22 from a population with standard deviation σ = 3.

(a) What is the value of c = 1 − α?


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
upper limit

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

Answer 1

The hypotheses with the significance level are

a) The value of c is

ans: c=0.95 (or 95% confidence interval)

We have the following sample information

n=33 is the sample size

is the sample mean

is the population standard deviation

is the standard error of mean

Since the sample size is greater than 30 (and also we know the population standard deviation) we can say that the sampling distribution of mean is normal. That is we will use z-test.

The critical values for c=0.95 confidence interval. The area under each tail is

the right tail critical value is found by

Using the standard normal tables we can get that for z=1.96, P(Z<1.96)=0.975. Hence the right tail critical value is

We can now find the 95% confidence interval for mean as

ans: A 95% confidence interval for mean is

lower limit	20.98
Upper limit	23.02

The null hypothesis is and hence k=21

ans: k=21

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 μ based on the sample data. When k falls within the c = 1 –  α confidence interval, we do not reject H₀.

Here, k=21 falls with in the confidence interval [20.98,23.02].

ans: We do not reject  H₀.