Question

1000 randomly selected Americans were asked if they believed the minimum wage should be raised. 600 said yes. Construct a 95% confidence interval for the proportion of Americans who believe that the minimum wage should be raised.

a. Write down the formula you intend to use with variable notation).
b. Write down the above formula with numeric values replacing the symbols.
c. Write down the confidence interval in interval notation.

Answer 1

a. p`± z₀.₀₂₅
`\sqrt{ (p`q`)/(n)`

b.0.6 ± 1.96
`\sqrt (0.6* 0.4)/(1000)`

c. { -1.96 ≤ p`± z₀.₀₂₅
`\sqrt{ (p`q`)/(n)` ≥ 1.96} = 0.95

Explanation:

Here the total number of trials is n= 1000

The number of successes is p` = 600/1000 = 0.6. The q` is 1 - p`= 1- 0.6 = 0.4

The degree of confidence is 95 % therefore z₀.₀₂₅ = 1.96 ( α/2 = 0.025)

a. The formula used will be

p`± z₀.₀₂₅
`\sqrt{ (p`q`)/(n)` ( z with the base alpha by 2 (α/2 = 0.025))

b. Putting the values

0.6 ± 1.96
`\sqrt (0.6* 0.4)/(1000)`

c. Confidence Interval in Interval Notation.

{ -1.96 ≤ p`± z₀.₀₂₅
`\sqrt{ (p`q`)/(n)` ≥ 1.96} = 0.95

{ -z( base alpha by 2) ≤ p`± z₀.₀₂₅
`\sqrt{ (p`q`)/(n)` ≥ z( base alpha by 2) } = 1- α