Hypothesis Test For Population Mean "\\mu".
Since "\\sigma" is unknown, then we use "t-"statistic. The random variable "{\\bar{X}-\\mu \\over s\/\\sqrt{n}}" has a Student's t-distribution with "df=n-1" degrees of freedom. We reject "H_0" if "t<-t_{\\alpha\/2;n-1}" or "t>t_{\\alpha\/2;n-1}".
Given that "\\alpha=0.01, \\mu=250, \\bar{X}=258, s=16." What is the value of n?
Let "n=25." Then "df=n-1=25-1=24." Define the critical value from the table
The rejection region for this two-tailed test is
Decision about the null hypothesis
Since it is observed that "|t|=2.500<2.797," it is then concluded that the null hypothesis is not rejected.
Conclusion
It is concluded that the null hypothesis "H_0" is not rejected. Therefore, there is not enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level.
From the analysis of the table of critical values we see that if we increase "n" (and the number of degrees of freedom) and don't change the value of "\\alpha", then the critical value decreases.
On the other hand if we increase "n" and don't change "\\bar{X}, \\mu, s" the random variable "{\\bar{X}-\\mu \\over s\/\\sqrt{n}}" increases.
Let "n=30." Then "df=n-1=30-1=29." Define the critical value from the table
The rejection region for this two-tailed test is
Decision about the null hypothesis
Since it is observed that "|t|=2.739<2.756," it is then concluded that the null hypothesis is not rejected.
Conclusion
It is concluded that the null hypothesis "H_0" is not rejected. Therefore, there is not enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level.
Let "n=31." Then "df=n-1=31-1=30." Define the critical value from the table
The rejection region for this two-tailed test is
Decision about the null hypothesis
Since it is observed that "|t|=2.784>2.750," it is then concluded that the null hypothesis is rejected.
Conclusion
It is concluded that the null hypothesis "H_0" is rejected. Therefore, there is enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level. Therefore, the machine needs adjustment.
If we take "n\\leq30" there is not enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level.
If we take "n\\geq31" there is enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level. Therefore, the machine needs adjustment.
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