Question

This problem demonstrates how to calculate the confidence interval for a population proportion. After, you will be asked to redo the calculations (with small variations).

A survey was conducted to determine how many people were in favor of a proposed law to criminalize texting while driving. Response options included strongly disagree, disagree, neither agree or disagree, agree, and strongly agree. The survey asked a random sample of 800 18- to 25-year-olds, and 648 indicated agreeing or strongly agreeing with the proposed law. Calculate the 98% confidence interval for the proportion of 18- to 25-year-olds in favor of the proposed law.

Step 1: Calculate the point estimate.
Calculate the sample proportion by dividing the number in favor by the sample size:

ˆp=kn=648800=0.81p^=kn=648800=0.81

Step 2: Calculate the estimate for the sampling distribution standard deviation.
The standard deviation of the sampling distribution for proportions (also called the standard error), is obtained with the formula

ˆσ=√ˆp(1−ˆp)n=√(0.81)(0.19)800=0.01387

Step 3: Calculate the critical value zα/2zα/2.
The 98% confidence interval spans the middle-most 98% of the normal distribution. This means there is 2% in the tails, and 1% in each individual tail. The critical values can be found using Excel:
      = invNorm(1%)
which gives the value
      -2.326348
The critical values would be the positive and negative of this value, or
      ±2.326348.

Step 4: Calculate the margin of error.
The margin of error is the critical value times the standard error:

ME=zα/2⋅ˆσ=(2.326348)(0.01387)=0.032266

Step 5: Calculate the confidence interval.
Add and subtract the margin of error from the point estimate to obtain the confidence interval:
      lower bound = ˆp−ME=0.81−0.032266=0.777734
and
      upper bound = ˆp+ME=0.81+0.032266=0.842266
Putting it all together, the 98% confidence interval is

77.8% < p < 84.2%

Repeat the above calculation, but this time calculate the 99% confidence interval.
      • critical value:  zα/2= 2.576
      • margin of error: ME = .0358
      • 99% confidence interval:  % < p < %

Repeat the initial calculation (with the original 98% confidence level), but this time the sample was of 900 adults and 729 were in favor (a sample with the same point estimate).
      • standard error:  σ^=  
      • margin of error: ME =
      • 98% confidence interval:  % < p < %

Answer 1

Solution:- Given that above sum = p = 0.81

=> 99% confidence interval for the population proportions = p +/- ME
= 0.81 +/- 0.0358
= 0.7742,0.8458

-------------------------------
Solution:- Given that n = 900, X = 729 , p = X/n = 729/900 = 0.81

q = 1-p = 0.19

standard error: σ^ = sqrt(pq/n) = sqrt(0.81*0.19/900) = 0.0131

Margin of error = Z*sqrt(pq/n) = 2.33*0.0131 = 0.0305

98% confidece interval : 0.81 +/- 0.0305 = (0.7795 , 0.8405)