Question

A simple random sample of 800 elements generates a sample proportion overline p =0.66 . ( Round your answers to four decimal places . ) ( a ) Provide a 90 % confidence interval for the population proportion . to ( b ) Provide a 95 % confidence interval for the population proportion . to

Answer 1

a. A 90% confidence interval for the population proportion is [0.6286, 0.6914].

b. A 95% confidence interval for the population proportion is [0.6226, 0.6974].

In Mathematics and Statistics, the sample proportion of a sample can be calculated by using this formula:

`\hat{p} = (x)/(n)`

Where:

• x represent the total number of individuals that are having a specified characteristic.
• n represent the total number of individuals that are in the sample.

Part a.

For a confidence level of 90%, the critical value of z is given by;

Critical value, z* = 1.645

Now, we can calculate the confidence interval (CI) by using the following formula;

`CI=\hat{p}\pm z^(*)\sqrt{\frac{\hat{p}(1-\hat{p})}{n} } \\\\CI=0.66 \pm 1.645 \sqrt{(0.66(1-0.66))/(800) } \\\\CI=0.66 \pm 1.645 \sqrt{(0.2904)/(800) }`

CI = [0.66 ± 1.645(0.0191)]

CI = [0.66 ± 0.0314]

CI = [0.66 - 0.0314, 0.66 + 0.0314]

CI = [0.6286, 0.6914]

Part b.

For a confidence level of 95%, the critical value of z is given by;

Critical value, z* = 1.960

Now, we can calculate the confidence interval (CI) by using the following formula;

`CI=\hat{p}\pm z^(*)\sqrt{\frac{\hat{p}(1-\hat{p})}{n} } \\\\CI=0.66 \pm 1.960 \sqrt{(0.66(1-0.66))/(800) } \\\\CI=0.66 \pm 1.960 \sqrt{(0.2904)/(800) }`

CI = [0.66 ± 1.960(0.0191)]

CI = [0.66 ± 0.0374]

CI = [0.66 - 0.0374, 0.66 + 0.0374]

CI = [0.6226, 0.6974]

Answer 2

STEP - BY - STEP EXPLANATION

Whatto find?

• 90% confidence interval for the population proportion.

,

• 95% confidence interval for the ppulation proportion.

Given:

n= 800

p=0.66

q=1- p= 1- 0.66 =0.34

a) Given a 90% confidence interval,

α =1- 0.90= 0.10

`Z_{(\alpha)/(2)}=Z_{(0.10)/(2)}=Z_(0.05)=1.645`
`C.I=\bar{P}\pm Z_{(\alpha)/(2)}*\sqrt{\frac{\bar{P}\bar{q}}{n}}`
`=0.66\pm1.645*\sqrt{(0.66*0.34)/(800)}`
`\begin{gathered} =0.66\pm0.02755 \\ \\ =(0.6324,\text{ 0.6276\rparen} \end{gathered}`

b)

Given 95% confidence interal.

1- 0.95 =0.

ANSWER

a) (0.6276, 0.6324)