Question

A high school principal wishes to estimate how well his students are doing in math. Using 40 randomly chosen tests, he finds that 77% of them received a passing grade. Create a 99% confidence interval for the population proportion of passing test scores. Enter the lower and upper bounds for the interval in the following boxes, respectively. You may answer using decimals rounded to four places or a percentage rounded to two. Make sure to use a percent sign if you answer using a percentage.

Answer 1

99% confidence interval for the population proportion of passing test scores is [0.5986 , 0.9414].

Explanation:

We are given that a high school principal wishes to estimate how well his students are doing in math.

Using 40 randomly chosen tests, he finds that 77% of them received a passing grade.

Firstly, the pivotal quantity for 99% confidence interval for the population proportion is given by;

P.Q. =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion of students received a passing grade = 77%

n = sample of tests = 40

p = population proportion

Here for constructing 99% confidence interval we have used One-sample z proportion test statistics.

So, 99% confidence interval for the population proportion, p is ;

P(-2.5758 < N(0,1) < 2.5758) = 0.99 {As the critical value of z at 0.5%

level of significance are -2.5758 & 2.5758}

P(-2.5758 <
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 2.5758) = 0.99

P(
`-2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` <
`{\hat p-p}` <
`2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.99

P(
`\hat p-2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` < p <
`\hat p+2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ) = 0.99

99% confidence interval for p = [
`\hat p-2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }` ,
`\hat p+2.5758 * {\sqrt{(\hat p(1-\hat p))/(n) } }`]

= [
`0.77-2.5758 * {\sqrt{(0.77(1-0.77))/(40) } }` ,
`0.77+2.5758 * {\sqrt{(0.77(1-0.77))/(40) } }` ]

= [0.5986 , 0.9414]

Therefore, 99% confidence interval for the population proportion of passing test scores is [0.5986 , 0.9414].

Lower bound of interval = 0.5986

Upper bound of interval = 0.9414