2.44 Prayer among the Millennials. The Millennial gen- eration (so called becaus

Question

2.44 Prayer among the Millennials. The Millennial gen- eration (so called because they were born after 1980 and began to come of age around the year 2000) are less religiously active than older Americans. One of tions in the General Social Survey in 2012 was, "How often does the respondent pray?" Among the 457 respondents in the survey between 18 and 32 years of age, 289 prayed at least once a week.0 the ques years of age, 289 prayed at least once a week.30 Assume that the sample is an SRS. (a) Verify that the sample size conditions are met for the large-sample confidence interval. What is Exploring the Web

Explanation / Answer

Answer a

Here, we are given n = 457 and x = 289

So, p = 289/457 = 0.632385

Two conditions np > 5 and nq > 5

First condition = np = 457*0.632385 = 289 > 5

Here, q = 1 – p = 1 – 0.632385 = 0.367615

Second condition = nq = 457*0.367615 = 168 > 5

Both conditions are proved for large sample size.

Also, sample is simple random sample.

Now, we have to find the 99% confidence interval.

The confidence interval is given as below:

Standard error = sqrt(pq/n) = sqrt(0.632385*0.367615/457) = 0.0226

Critical value = 2.5758

Lower limit = sample proportion – critical value*standard error

Lower limit = 0.63238512 – 2.5758*0.0226

Lower limit = 0.5743

Lower limit = sample proportion + critical value*standard error

Upper limit = 0.63238512 + 2.5758*0.0226

Upper limit = 0.6905

Answer b

Here, we have to find plus four confidence interval

Here, p = (x+2) / (n+4) = (289 + 2) / (457 + 4) = 291 / 461 = 0.631236

So, q = 0.368764

Standard error = sqrt(0.631236*0.368764/457) = 0.022569

Critical value = 2.5758

Lower limit = sample proportion – critical value*standard error

Lower limit = 0.631236 – 2.5758*0.022569

Lower limit = 0.573103

Lower limit = sample proportion + critical value*standard error

Upper limit = 0.631236 + 2.5758*0.022569

Upper limit = 0.68937

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