According to a study conducted by a statistical organization, the proportion of

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According to a study conducted by a statistical organization, the proportion of people who are satisfied with the way things are going in their lives is 0.80. Suppose that a random sample of 100 people is obtained. Complete parts (a) nough(e) beluw (a) Suppose the random sample of 100 people is asked. "Are you satisfied with the way things are going in your life?" Is the response to this question qualitative or quantitative? Explain OA. The response is qualitative because the responses can be classified based on the characteristic of being satisfied or not. B. The response is quantitative because the responses can be classified based on the characteristic of being satisfied or not. O C. The response is quantitative bocause the rosponses can be measured numorically and tho values added or subtractod, providing meaningful rosults O D. The rocponce is qualitativo bocause the rocponeoe can bo meacurod numorically and the valuoe addod or cubtractod, providing moaningful recults. (b) Explain why the sample proportion. p. is a random variable. What is the source of the variability? O A. The sample proportion p io a random variable because the value of prepresents a random person included in the sample. The variability is due to the fact that different people feel differently regarding their satisfaction. D. The sample proport on p s a random vanable because he value o p vanes rom sample to sample The va a y is due o ne that peo e may not be respon in the quest on t OC. The sample proportion p is a random variable because the value of p represents a random person included in the sample. The variability is due to the fact that people may not be responding to the question truthfully. O D. The sample propotion p is a random variable because the value of p varies from sample to sample. The variability is due to the fact that diferent people feel differenly regarding their satisfaction (c) Describe the sampling distribution of p, the proportion of people who are satisfied with the way things are going in their life. Be sure to verify the model requirements. Since the sample size is no than 5% of the population size and np(1-p- 210, the distribution of p is Click lu selecl yuur aswes). | with ^-| | and ^-

Explanation / Answer

a. The variable measured is categorical in nature, or qualitative as the variable can be classified into two categories-being satisfied or not. Option B is thus not valid. Options C is incorrect as, the response variable in question cannot be measured numerically followed by mathematical operations to be performed on it. Option D is also not meaningful as mathematical operations can not be performed on qualitative variable. Thus, right answer is A.

b.A random sample of 100 people is obtained which implies that the sample proportion phat varies from sample to sample. The variation might be due to different people feeling differently regarding their satisfaction. The first part of option B is correct, but the second part is not right. As the question asked is apolitical, hence it can be assumed respondents will be giving spontaneous response. Options A and C are also rejected. Option D is correct.

c. The sample size 100 is no more than 5% of the population size and np(1-p)=100(0.80)(1-0.80)=16, the distribution of phat is normal with muphat=0.80 and sigmaphat=sqrt [p(1-p)/n]=sqrt [0.80(1-0.80)/100]=0.04

d. Assume X denote probability of proportion who are satisfied with the way things are going in their life.

P(X>0.83)=P[X>(0.83-0.80)/0.04] [substitute values in z score formula, z=(X-muphat)/sigma phat]

=P(X>0.75) (see z table formula to find the area to the left of z=0.75 and then subtract it from 1 to find area beyond it)

=1-0.7734

=0.2266 (ans)

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