Hypothesis Test For Population Mean µ
"H_0:\\mu=\\mu_0=250""H_1: \\mu\\not=\\mu_0 (two-tailed \\ test)"
Hypothesis Test When σ Is Not Known:
The random variable
"{\\bar{X}-\\mu_0 \\over s\/\\sqrt{n}}"has a Student's t-distribution with n-1 degrees of freedom. Reject "H_0" if "t<-t_{\\alpha\/2;n-1}" or "t>t_{\\alpha\/2;n-1}".
Given that "\\alpha=0.01, \\mu_0=250, \\bar{X}=258, s=16"
Let "n=25." Then "df=n-1=24."
The rejection region for this two-tailed test is
Let "n=30." Then "df=30-1=29."
The rejection region for this two-tailed test is
Let "n=31." Then "df=n-1=30."
The rejection region for this two-tailed test is
Let "n=36." Then "df=n-1=35."
The rejection region for this two-tailed test is
Let "n=49." Then "df=49-1=48."
The rejection region for this two-tailed test is
The t-statistic is computed as follows:
If "n=25"
Decision about the null hypothesis
Since it is observed that "|t|=2.5<2.796940," it is then concluded that the null hypothesis is not rejected.
Conclusion
It is concluded that the null hypothesis Ho 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.
If "n=30."
Decision about the null hypothesis
Since it is observed that "|t|=2.738613<2.756386," it is then concluded that the null hypothesis is not rejected.
Conclusion
It is concluded that the null hypothesis Ho 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.
If "n=31."
Decision about the null hypothesis
Since it is observed that "|t|=2.783882>2.749996," it is then concluded that the null hypothesis is rejected.
Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level.
If "n=36"
Decision about the null hypothesis
Since it is observed that "|t|=3>2.723806," it is then concluded that the null hypothesis is rejected.
Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean "\\mu" is different than 250, at the 0.01 significance level.
We see that if "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.
Comments
c) One may assume a sample of cartons is not very large in practice, for example, less than 50. In general, it is possible to apply the sample size greater than 49.
a) The value 2.796940 was taken from the table of critical values for t-distribution (the row 24 and the column 0.005) at https://www.thoughtco.com/student-t-distribution-table-3126265 or with a help of the command qt(1-.01/2,24) in R Studio; b) The sample size was not given in the question, hence different values of the sample size were used in the solution.
How you perform a calculation of a) t α/2;n−1 =t 0.01/2;25−1=2.796940 b) How you can decide n=25 and based on what assumption. c) Why stop until n=49?
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