solution:
The critical value approach involves determining "likely" or
"unlikely" by determining whether or not the observed test
statistic is more extreme than would be expected if the null
hypothesis were true. That is, it entails comparing the observed
test statistic to some cutoff value, called the "critical
value." If the test statistic is more extreme than the
critical value, then the null hypothesis is rejected in favor of
the alternative hypothesis. If the test statistic is not as extreme
as the critical value, then the null hypothesis is not
rejected.
the four steps involved in using the critical value approach to
conducting any hypothesis test are
- Specify the null and alternative hypotheses.
- Using the sample data and assuming the null hypothesis is true,
calculate the value of the test statistic. To conduct the
hypothesis test for the population mean μ, we use the
t-statistic t∗=x¯−μs/n which follows a
t-distribution with n - 1 degrees of
freedom.
- Determine the critical value by finding the value of the known
distribution of the test statistic such that the probability of
making a Type I error — which is denoted α (greek letter "alpha")
and is called the "significance level of the test"
— is small (typically 0.01, 0.05, or 0.10).
- Compare the test statistic to the critical value. If the test
statistic is more extreme in the direction of the alternative than
the critical value, reject the null hypothesis in favor of the
alternative hypothesis. If the test statistic is less extreme than
the critical value, do not reject the null hypothesis .In our
example concerning the mean grade point average, suppose we take a
random sample of n = 15 students majoring in mathematics.
Since n = 15, our test statistic t* has
n - 1 = 14 degrees of freedom. Also, suppose we set our
significance level α at 0.05, so that we have only a 5% chance of
making a Type I error he critical value for conducting the
right-tailed test H0 :
μ = 3 versus HA : μ > 3 is
the t-value, denoted tα, n -
1, such that the probability to the right of it is α. It
can be shown using either statistical software or a
t-table that the critical value t
0.05,14 is 1.7613. That is, we would reject the null
hypothesis H0 : μ = 3 in favor of the
alternative hypothesis HA : μ > 3
if the test statistic t* is greater than 1.7613