Given the following information, construct the rejection region. Show the solution in
a step-by-step procedure.
H 0 : = 45
H a : < 45
= 40, = 12, n = 32, = 0.01
The following null and alternative hypotheses need to be tested:
"H_0:\\mu=45"
"H_a:\\mu\\not=45"
This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.
Based on the information provided, the significance level is "\\alpha = 0.01," "df=n-1=31" degrees of freedom, and the critical value for a two-tailed test is "t_c =2.744042."The rejection region for this two-tailed test is "R = \\{t:|t|> 2.744042\\}."
The t-statistic is computed as follows:
Since it is observed that "|t| = 2.357<2.744042=t_c," it is then concluded that the null hypothesis is not rejected.
Using the P-value approach:
The p-value for two-tailed "df=31" degrees of freedom, "t=-2.357" is "p=0.024919," and since "p=0.024919>0.01=\\alpha," it is concluded that the null hypothesis is not rejected.
Therefore, there is not enough evidence to claim that the population mean "\\mu" is different than 45, at the "\\alpha = 0.01" significance level.
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