Question

Although newspapers frequently report survey results, they typically do not use confidence intervals.  Instead, they will report the population mean with a plus/minus amount of error (e.g. 25% ± 2%).  For the following survey information, convert the “plus/minus” language into a confidence interval, and then, using the “Confidence Intervals for Population Proportion” spreadsheet, approximate the value of the level of confidence, c, for each situation. In a survey of 8451 U.S. adults, 31.4% said they were taking vitamin E as a supplement.  The survey’s margin of error is plus or minus 1%.In a survey of 1001 U.S. adults, 27% said they had smoked a cigarette in the past week.  The survey’s margin of error is plus or minus 3%.

Answer 1

1) Confidence Interval = (31.4% ± 1%)

Confidence Interval = (30.4%, 32.4%) = (0.304, 0.324)

Level of confidence = 95%

2) Confidence Interval = (27% ± 3%)

Confidence Interval = (24%, 30%) = (0.24, 0.30)

Level of confidence = 97%

Note that the z-table chart was used to match the critical value to level of confidence.

Step-by-step explanation

Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample proportion) ± (Margin of error)

So, we first calculate the confidence interval for each of these questions

1) Sample proportion = 31.4% = 0.314

Margin of error = 1% = 0.01

Confidence Interval = (Sample proportion) ± (Margin of error)

Confidence Interval = 31.4% ± 1%

Confidence Interval = (30.4%, 32.4%) = (0.304, 0.324)

2) Sample proportion = 27% = 0.27

Margin of error = 3% = 0.03

Confidence Interval = (Sample proportion) ± (Margin of error)

Confidence Interval = 27% ± 3%

Confidence Interval = (24%, 30%) = (0.24, 0.30)

Part B

To find the level of confidence for each of them, we need to first define the Margin of error and subsequently, the critical value, which will directly give the level of confidence from the table.

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error)

Standard error of the mean = σₓ = √(pq/n)

Question 1

p = sample proportion = 0.314

q = 1 - p = 1 - 0.314 = 0.686

n = sample size = 8451

σₓ = √(0.314×0.686/8451) = 0.00505

Margin of Error = (Critical value) × (standard Error)

0.01 = (Critical value) × 0.00505

Critical value = (0.01/0.0505) = 1.98

From the tables,

when critical value = 1.98,

Confidence level = 95.15% = 95% to the nearest whole number.

Question 2

p = sample proportion = 0.27

q = 1 - p = 1 - 0.27 = 0.73

n = sample size = 1001

σₓ = √(0.27×0.73/1001) = 0.01403

Margin of Error = (Critical value) × (standard Error)

0.03 = (Critical value) × 0.01403

Critical value = (0.03/0.01403) = 2.143

From the tables,

when critical value = 2.143,

Confidence level = 96.95% = 97% to the nearest whole number.

Hope this Helps!!!