Answer to Question #342564 in Statistics and Probability for nana

Question #342564

(a) Identify the claim and state H0 and Ha. (b) Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test, a t-test, or a chi-square test. Explain your reasoning. (c) Choose one of the options. If convenient, use technology. Option 1: Find the critical value(s), identify the rejection region(s), and find the appropriate standardized test statistic. Option 2: Find the appropriate standardized test statistic and the P-value. (d) Decide whether to reject or fail to reject the null hypothesis. (e) Interpret the decision in the context of the original claim.


7. A researcher claims that the mean age of the residents of a small town is more than 32 years. The ages (in years) of a random sample of 36 residents are listed below. At a = 0.10, is there enough evidence to support the researcher’s claim? Assume the population standard deviation is 9 years.

41 33 47 31 26 39 19 25 23 31 39 36

41 28 33 41 44 40 30 29 46 42 53 21

29 43 46 39 35 33 42 35 43 35 24 21


1
Expert's answer
2022-05-24T17:57:18-0400
"\\bar{X}=\\dfrac{1}{36}(41+33+47+31+26+39"

"+19+25+23+31+39+36+41+ 28"

"+ 33+ 41+ 44 +40+ 30+ 29+ 46+ 42"

"+ 53+ 21+ 29+ 43 +46 +39 +35 +33"

"+ 42+ 35+ 43 +35+ 24+ 21)=\\dfrac{421}{12}\\approx35.0833"

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

"H_0:\\mu\\le32"

"H_1:\\mu>32"


(b) corresponds to a right-tailed one-tailed test, for which a z-test for one mean, with known population standard deviation will be used.


(c)Based on the information provided, the significance level is "\\alpha = 0.10," and the critical value for a right-tailed test is "z_c =1.2816."

The rejection region for this right-tailed test is "R = \\{z:z>1.2816\\}."

The z-statistic is computed as follows:



"z=\\dfrac{\\bar{x}-\\mu}{\\sigma\/\\sqrt{n}}=\\dfrac{\\dfrac{421}{12}-32}{9\/\\sqrt{36}}\\approx2.0556"


(d) it is observed that "z=2.0556>1.2816=z_c," it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is "p=P(z>2.0556)=0.019911," and since "p= 0.019911<0.10=\\alpha," it is concluded that the null hypothesis is rejected.


(e) Therefore, there is enough evidence to claim that the population mean "\\mu"

is greater than 32, at the "\\alpha = 0.10" significance level.



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