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Talja [164]
2 years ago
6

Customers arrive at a movie theater at the advertised movie time only to find that they have to sit through several previews and

prepreview ads before the movie starts. Many complain that the time devoted to previews is too long (The Wall Street Journal, October 12, 2012). A preliminary sample conducted by The Wall Street Journal showed that the standard deviation of the amount of time devoted to previews was four minutes. Use that as a planning value for the standard deviation in answering the following questions.
a. If we want to estimate the population mean time for previews at movie theaters with a margin of error of 72 seconds, what sample size should be used? Assume 95% confidence.
b. If we want to estimate the population mean time for previews at movie theaters with a margin of error of 1 minute, what sample size should be used? Assume 95% confidence.
Mathematics
1 answer:
balu736 [363]2 years ago
3 0

Answer:

(a) The sample size required is 43.

(b) The sample size required is 62.

Step-by-step explanation:

The (1 - α) % confidence interval for population mean is:

 CI=\bar x\pm z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}

The margin of error for this interval is:

 MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}

(a)

The information provided is:

<em>σ</em> = 4 minutes

MOE = 72 seconds = 1.2 minutes

Confidence level = 95%

α = 5%

Compute the critical value of z for α = 5% as follows:

 z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96

*Use a z-table.

Compute the sample size required as follows:

 MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}

       n=[\frac{z_{\alpha/2}\times \sigma}{MOE} ]^{2}

          =[\frac{1.96\times 4}{1.2}]^{2}\\\\=42.684\\\\\approx 43

Thus, the sample size required is 43.

(b)

The information provided is:

<em>σ</em> = 4 minutes

MOE = 1 minute

Confidence level = 95%

α = 5%

Compute the critical value of z for α = 5% as follows:

 z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96

*Use a z-table.

Compute the sample size required as follows:

 MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}

       n=[\frac{z_{\alpha/2}\times \sigma}{MOE} ]^{2}

          =[\frac{1.96\times 4}{1}]^{2}\\\\=61.4656\\\\\approx 62

Thus, the sample size required is 62.

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Population estimates, July 1, 2019, (V2019) 6,829,174

Population estimates base, April 1, 2010, (V2019) 6,346,276

Population, percent change - April 1, 2010 (estimates base) to July 1, 2019, (V2019) 7.6%

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Persons under 18 years, percent 22.1%

Persons 65 years and over, percent 16.7%

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Black or African American alone, percent(a) 17.1%

American Indian and Alaska Native alone, percent(a) 0.5%

Asian alone, percent(a) 2.0%

Native Hawaiian and Other Pacific Islander alone, percent(a) 0.1%

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Foreign born persons, percent, 2015-2019 5.1%

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Owner-occupied housing unit rate, 2015-2019 66.3%

Median value of owner-occupied housing units, 2015-2019 $167,200

Median selected monthly owner costs -with a mortgage, 2015-2019 $1,244

Median selected monthly owner costs -without a mortgage, 2015-2019 $388

Median gross rent, 2015-2019 $869

Building permits, 2019 41,361

Families & Living Arrangements  

Households, 2015-2019 2,597,292

Persons per household, 2015-2019 2.52

Living in same house 1 year ago, percent of persons age 1 year+, 2015-2019 85.5%

Language other than English spoken at home, percent of persons age 5 years+, 2015-2019 7.2%

Computer and Internet Use  

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Households with a broadband Internet subscription, percent, 2015-2019 78.4%

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Health  

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In civilian labor force, female, percent of population age 16 years+, 2015-2019 56.6%

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