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Losing weight A Gallup Poll in November 2008found that 59%of the people in its sample said 鈥淵es鈥 when asked, 鈥淲ould you like to lose weight?鈥 Gallup announced: 鈥淔or results based on the total sample of national adults, one can say with 95% confidence that the margin of (sampling) error is 3percentage points.鈥

(a) Explain what the margin of error means in this setting.

(b) State and interpret the 95% confidence interval.

(c) Interpret the confidence level.

Short Answer

Expert verified

a). on average the sample proportion will be within 3%of the true proportion in 95%of all samples.

b). We are 95%confident that the true population proportion is between 0.56and 0.62.

c). On average, in 95% of all samples the corresponding confidence interval will contain the true population proportion p.

Step by step solution

01

Part (a) Step 1: Given Information 

For the total sample of national adults, one can say with 95%confidence that the margin of (sampling) error is 3 percentage points.

02

Part (a) Step 2: Explanation

The margin of error is the amount of error that is made, on average, in estimating the population proportion p by the sample proportion p^.

Thus on average the sample proportion will be within 3% of the true proportion in 95% of all samples.

03

Part (b) Step 1: Given Information 

p^=59%=0.59

04

Part (b) Step 2: Explanation 

The boundaries of the confidence interval is the sample proportion increased/decreased by the margin of error:

0.56=0.59-0.03=p^-E<p<p^+E=0.59+0.03=0.62

We are 95%confident that the true population proportion is between 0.56and 0.62.

05

Part (c) Step 1: Given Information

The confidence level is 95%.

06

Part (c) Step 2: Explanation 

The confidence level is 95%.

This means: On average, in 95%of all samples the corresponding confidence interval will contain the true population proportion p.

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