A hat contains 10,000 balls. A random sample of 100 balls is selected from the hat....

A hat contains 10,000 balls. A random sample of 100 balls is selected from the hat. 17 of the selected balls are red, 27 are green and 56 are black. Construct a 95% confidence interval estimate for the percentage of green balls in the hat.

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Expert Solution

Solution :

Given that,

n = 100

x = 27

Point estimate = sample proportion = $\hat p$ = x / n = 27 / 100 = 0.270

1 - $\hat p$ = 1 - 0.270 = 0.730

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.96

Margin of error = E = Z_{/ 2} * $\sqrt$ (( $\hat p$ * (1 - $\hat p$ )) / n)

= 1.96 * ( $\sqrt$ ((0.270 * 0.730) / 100)

= 0.087

A 95% confidence interval for population proportion p is ,

$\hat p$ - E < p < $\hat p$ + E

0.270 - 0.087 < p < 0.270 + 0.087

0.183 < p < 0.357

a 95% confidence interval estimate for the percentage of green balls in the hat is : 18.3% to 35.7%

orchestra answered 1 year ago

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