   Chapter 6, Problem 6.16P

Chapter
Section
Textbook Problem

S O C The World Values Survey is administered periodically to random samples from societies around the globe. Listed here are the percentage of respondents in each nation who said that they are “very happy.” Construct confidence interval estimates for each nation at the 95% level. Nation Percent“Very happy” Sample Size Confidence Intreval Germany 21.1% 2046 Japan 32.3% 2443 Chili 24.4% 1000 China 15.7% 2300 Malaysia 56.6% 1300 Nigeria 55.6% 1759 Russia 14.8% 2200

To determine

To find:

The confidence interval for the population mean for the specified level.

Explanation

Given:

The given information is,

The World Values Survey is administered periodically to random samples from societies around the globe.

The given table of information is,

 Nation Percent“Very happy” Sample Size Confidence Intreval Germany 21.1% 2046 Japan 32.3% 2443 Chili 24.4% 1000 China 15.7% 2300 Malaysia 56.6% 1300 Nigeria 55.6% 1759 Russia 14.8% 2200

Approach:

The confidence interval is an interval estimate from the statistics of the observed data that might contain the true value of the unknown population parameter.

Formula used:

For large samples and given sample proportion, the confidence interval (c.i.) is given by,

c.i.=Ps±Z(Pu(1Pu)N)

Where, Ps is the sample proportion,

Pu is the population proportion,

Z is the Z score determined by the alpha level and,

N is the sample size.

Calculation:

The level of confidence is 95%.

For 95%, the Z value is given as,

Z(95%)=1.96

For large samples and given sample proportion, the confidence interval (c.i.) is given by,

c.i.=Ps±Z(Pu(1Pu)N)……(1)

The confidence interval for Germany is given as,

Substitute 0.211 for Ps, 0.5 for Pu, 1.96 for Z and 2046 for N in equation(1) to obtain the confidence interval.

c.i.=0.211±1.96(0.5×0.52046)c.i.=0.211±0.022

Thus, the 95% confidence interval for Germany is,

c.i.=0.211±0.022

The confidence interval for Japan is given as,

Substitute 0.323 for Ps, 0.5 for Pu, 1.96 for Z and 2443 for N in equation(1) to obtain the confidence interval.

c.i.=0.323±1.96(0.5×0.52443)c.i.=0.323±0.02

Thus, the 95% confidence interval for Japan is,

c.i.=0.323±0.02

The confidence interval for Chili is given as,

Substitute 0.244 for Ps, 0.5 for Pu, 1.96 for Z and 1000 for N in equation(1) to obtain the confidence interval.

c.i.=0.244±1.96(0.5×0.51000)c.i.=0.244±0.031

Thus, the 95% confidence interval for Chili is,

c.i.=0.244±0.031

The confidence interval for China is given as,

Substitute 0.157 for Ps, 0.5 for Pu, 1.96 for Z and 2300 for N in equation(1) to obtain the confidence interval.

c.i

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