For a two-tailed hypothesis test at the significance H0:u=u0 will be rejected in favor of the alternative hypothesis Ha:u‰ u0 if and only if u0...

We are asked
to find the interval in which we would be unable to reject the null hypothesis.

When hypothesis testing we generally state the null hypothesis, the alternative
hypothesis, and the claim. Then we find the critical values; these values delineate the regions
in which we will reject the null-hypothesis (the critical region(s)) and the region where we
fail to reject the null-hypothesis (the non-critical region). We then use the data to find the
test value, determine if the test value is in the critical region and decide whether to reject
the null-hypothesis or not.

1. `H_{0}:mu=mu_{0}`

`H_{1}:mu
ne mu_{0}`

Here `H_{0}: mu=4.8`

`H_{1}: mu ne 4.8` (This
is the claim.)

2. We now find the critical values. Note that we are doing a
two-tailed test. (If we were checking only less than or greater than, it would be a one-tail
test.) Since we are using data from a sample, in particular we do not know the population
standard deviation, we will use the Student's T-table to find the critical values.

Since the size of the sample is n=20, the degrees of freedom are 19. We are running a
two-tailed test with `alpha=.05` so we want a 95% confidence.

From the table
we find the critical values to be `+- 2.093`

This is denoted `t_{alpha/2}`
.

3. The test value is found by
`t=(bar(x)-mu)/(s/sqrt(n))=(4.33-4.8)/(2.247/sqrt(20))~~-0.935`

Since the
test value is within the noncritical region we would not reject the null-hypothesis.

Essentially, the confidence interval about the mean of the sample will give the
boundaries of the noncritical region.

The confidence interval is given
by:

`bar(x)-t_{alpha/2}(s/sqrt(n))

`4.33-2.093(2.247/sqrt(20))

Thus the confidence interval is
`3.2784

Any value in this range will not allow us to reject
the null-hypothesis.

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