Question

1. A sample of 100 results in 27 successes.
a. Calculate the point estimate for the population proportion of successes. (Do not round intermediate calculations. Round your answer to 3 decimal places.)  

b. Construct 95% and 90% confidence intervals for the population proportion. (Round intermediate calculations to at least 4 decimal places. Round "z" value and final answers to 3 decimal places.)

95% -

90%-

c. Can we conclude at 95% confidence that the population proportion differs from 0.330?  

• No, since the confidence interval does not contain the value 0.330.

• No, since the confidence interval contains the value 0.330.

• Yes, since the confidence interval does not contain the value 0.330.

• Yes, since the confidence interval contains the value 0.330.

d. Can we conclude at 90% confidence that the population proportion differs from 0.330?

• No, since the confidence interval contains the value 0.330.

• No, since the confidence interval does not contain the value 0.330.

• Yes, since the confidence interval contains the value 0.330.

• Yes, since the confidence interval does not contain the value 0.330.

Answer 1

Solution :

Given that,

n = 100

x = 27

a)

Point estimate = sample proportion = = x / n = 27 / 100 = 0.270

1 - = 1 - 0.270 = 0.730

b)

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z _0.05 = 1.645

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 * (((0.270 * 0.730) / 100)

= 0.0730

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.270 - 0.0730 < p < 0.270 + 0.0730

0.1970 < p < 0.3430

(0.1970 , 0.3430)

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} * (( * (1 - )) / n)

= 1.96 * (((0.270 * 0.730) / 100)

= 0.0870

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.270 - 0.0870 < p < 0.270 + 0.0870

0.1830< p < 0.3570

(0.1830 , 0.3570)

c)

Yes, since the confidence interval contains the value 0.330

d)

Yes, since the confidence interval contains the value 0.330