Question

Sample proportion of .14 and standard deviation of.02, use empirical rule to construct a 95% confidence interval

Answer 1

The empirical rule states that 65% of the data under the normal curve is within 1 standard deviation of the mean, 95% of the data is within 2 standard deviations of the mean, and 99% is within 3 standard deviations of the mean.

The approximation to the distribution of the sample proportion has the following shape:

`\hat{p}\approx(p;(p(1-p))/(n))`

The mean of the distribution is the sample proportion: μ= p

The standard deviation of the distribution is the square root of the variance

σ=√[p(1-p)/n]

For the given distribution:

μ= 0.14

σ= 0.02

95% of the distribution is μ ± 2σ

Upper bound:

`\mu+2\sigma=0.14+2\cdot0.02=0.18`

Lower bound:

`\mu-2\sigma=0.14-2\cdot0.02=0.10`

The 95% confidence interval is [0.10;0.18]