Answer to Question #234845 in Statistics and Probability for liche

Question #234845

A random sample of 20 business statistics II students gave a summary of the information

on how they spend their time studying in hours, in the week before the busines statistics

II nal examinations: sample mean = 40 hours, sample standard deviation is 15. It is

assumed that the study time follows an approximate normal distribution.

(a) Test at the 5% signi cance level the null hypothesis that the population mean is

not more than 50 hours.

(b) Test at the 2:5% signi cance level the null hypothesis that the varaince is atleast

110 hours.

(c) Construct 95% CI for  and 2


1
Expert's answer
2021-09-09T13:23:58-0400

(a) The following null and alternative hypotheses need to be tested:

"H_0:\\mu\\leq50"

"H_1:\\mu>50"

This corresponds to a right-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=0.05," and the critical value for a right-tailed test "df=n-1=20-1=19" degrees of freedom is "t_c=1.729133."

The rejection region for this right-tailed test is "R=\\{t:t>1.729133\\}."

The t-statistic is computed as follows:


"t=\\dfrac{\\bar{x}-\\mu}{s\/\\sqrt{n}}=\\dfrac{40-50}{15\/\\sqrt{20}}\\approx-2.981424"

Since it is observed that "t=-2.981424<1.729133=t_c," it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value for right-tailed, "\\alpha=0.05, df=19, t=-2.981424" is

"p=0.996165," and since "p=0.996165>0.05=\\alpha," it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean "\\mu" is not more than 50 hours, at the "\\alpha=0.05" significance level.


(b) The following null and alternative hypotheses need to be tested:

"H_0: \\sigma^2\\geq110"

"H_1:\\sigma^2<110"

This corresponds to a left-tailed test test, for which a Chi-Square test for one population variance will be used. Based on the information provided, the significance level is "\\alpha=0.025,"  "df=n-1=20-1=19" degrees of freedom is "\\chi_c^2=8.9065."

The rejection region for this left-tailed test is "R=\\{\\chi^2: \\chi^2<8.9065\\}."  

The Chi-Squared statistic is computed as follows:


"\\chi^2=\\dfrac{(n-1)s^2}{\\sigma^2}=\\dfrac{(20-1)(15)^2}{110}\\approx38.8636"

Since it is observed that "\\chi^2=38.8636>8.9065=\\chi_c^2," it is then concluded that the null hypothesis is not rejected.

Therefore, there is enough evidence to claim that the population variance "\\sigma^2"  is at least

"110 (hours)^2," at the 0.025 significance level.


(c) The critical value for "\\alpha=0.05" and "df=n-1=20-1=19" degrees of freedom is "t_c=z_{1-\\alpha\/2;n-1}=2.093024."

The corresponding confidence interval is computed as shown below:


"CI=(\\bar{x}-t_c\\times\\dfrac{s}{\\sqrt{n}}, \\bar{x}+t_c\\times\\dfrac{s}{\\sqrt{n}})"

"=(40-2.093024\\times\\dfrac{15}{\\sqrt{20}}, 40+2.093024\\times\\dfrac{15}{\\sqrt{20}})"

"=(32.9798, 47.0202)"

Therefore, based on the data provided, the 95% confidence interval for the population mean is "32.9798<\\mu<47.0202," which indicates that we are 95% confident that the true population mean "\\mu" is contained by the interval "(32.9798, 47.0202)."


The critical values for "\\alpha=0.05" and "df=19" degrees of freedom are "\\chi_L^2=\\chi_{1-\\alpha\/2, n-1}=8.9065" and "\\chi_U^2=\\chi_{\\alpha\/2, n-1}=32.8523." The corresponding confidence interval is computed as shown below:


"CI(Variance)=(\\dfrac{(n-1)s^2}{\\chi_{\\alpha\/2, n-1}}, \\dfrac{(n-1)s^2}{\\chi_{1-\\alpha\/2, n-1}})"

"=(\\dfrac{19(15)^2}{32.9798}, \\dfrac{19(15)^2}{8.9065})\\approx(129.6248,479.9865)"

Therefore, based on the data provided, the 95% confidence interval for the population variance is "(129.6248,479.9865)."



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