Question

A certain set of plants were constantly dying in the dry environment that was provided. The plants were moved to a more humid environment where life would improve.

(a)

Before moving all of the plants, the researchers wanted to be sure the new environment was promoting life. The study found that 24 out of 40 of the plants were alive after the first month. What is the point estimate?

(b)

What is the 95% confidence interval for the population proportion? (Use a table or technology. Round your answers to three decimal places.)

,

(c)

What is the 99% confidence interval for the population proportion? (Use a table or technology. Round your answers to three decimal places.)

,

Answer 1

Solution :

Given that,

n = 40

x = 24

Point estimate = sample proportion = $\hat p$ = x / n = 24/40=0.6

1 - $\hat p$ = 1-0.6=0.4

At 95% confidence level

$\alpha$ = 1 - 95%  

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_{0.025 = 1.96}

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

= 1.96 ($\sqrt$((0.6*0.4) / 40)

= 0.152

A 95% confidence interval for population proportion p is ,

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

0.6-0.152 < p < 0.6+0.152

0.448< p < 0.752

(0.448,0.752)

b.

At 99% confidence level the z is ,

$\alpha$  = 1 - 99% = 1 - 0.99 = 0.01

$\alpha$ / 2 = 0.01 / 2 = 0.005

Z_$\alpha$/2 = Z_0.005 = 2.576

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

= 2.576 ($\sqrt$((0.6*0.4) / 40)

= 0.200

A 95% confidence interval for population proportion p is ,

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

0.6-0.200 < p < 0.6+0.200

0.400< p < 0.800

(0.400,0.800)