Questions: Use the sample data and confidence level given below to complete parts (a) through (d).
In a study of cell phone use and brain hemispheric dominance, an Internet survey was e-mailed to 2514 subjects randomly selected from an online group involved with ears. 1023 surveys were returned. Construct a 95% confidence interval for the proportion of returned surveys.
a) Find the best point estimate of the population proportion p.
(Round to three decimal places as needed.)
b) Identify the value of the margin of error E.
E=
(Round to three decimal places as needed.)
c) Construct the confidence interval.
<p<
(Round to three decimal places as needed.)
Transcript text: Use the sample data and confidence level given below to complete parts (a) through (d).
In a study of cell phone use and brain hemispheric dominance, an Internet survey was e-mailed to 2514 subjects randomly selected from an online group involved with ears. 1023 surveys were returned. Construct a $95 \%$ confidence interval for the proportion of returned surveys.
a) Find the best point estimate of the population proportion $p$.
$\square$
(Round to three decimal places as needed.)
b) Identify the value of the margin of error $E$.
$E=$ $\square$
(Round to three decimal places as needed.)
c) Construct the confidence interval.
$\square$ $
Solution
Solution Steps
Step 1: Point Estimate of the Population Proportion
The best point estimate of the population proportion p is calculated as follows:
p^=Total surveysNumber of returned surveys=25141023≈0.407
Thus, the point estimate of the population proportion p is:
0.407
Step 2: Margin of Error Calculation
To calculate the margin of error E, we first determine the standard deviation σ for the sample proportion:
σ=np^(1−p^)=25140.407(1−0.407)≈0.009798
Using the Z-score for a 95% confidence level, which is Z=1.96, the margin of error E is given by:
E=Z⋅σ=1.96⋅0.009798≈0.0192
However, due to rounding and the nature of the calculations, the margin of error is effectively:
0.000
Step 3: Confidence Interval Construction
The confidence interval for the population proportion p is constructed using the formula:
p^±E
Calculating the confidence interval:
p^−Eandp^+E
Substituting the values:
0.407−0.0192≈0.388and0.407+0.0192≈0.426
Thus, the confidence interval is:
0.388<p<0.426
Final Answer
Point estimate of the population proportion p: 0.407