Question

A sample of 120 results in 30 successes. [You may find it useful to reference the z table.]   

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.)  

confidence intervals 95%

Confidence intervals 90%

c. Can we conclude at 95% confidence that the population proportion differs from 0.310?  

• No, since the confidence interval does not contain the value 0.310.

• No, since the confidence interval contains the value 0.310.

• Yes, since the confidence interval does not contain the value 0.310.

• Yes, since the confidence interval contains the value 0.310.

d. Can we conclude at 90% confidence that the population proportion differs from 0.310?

• No, since the confidence interval contains the value 0.310.

• No, since the confidence interval does not contain the value 0.310.

• Yes, since the confidence interval contains the value 0.310.

• Yes, since the confidence interval does not contain the value 0.310.

Answer 1

Solution :

Given that,

n = 120

x = 30

Point estimate = sample proportion = = x / n = 30 / 120 = 0.250

1 - = 0.750

(a) At 95%

Z_/2 = Z _0.025 = 1.96

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

= 1.96 * (((0.250 * 0.750) / 120)

= 0.077

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.250 - 0.077< p < 0.250 + 0.077

0.173 < p < 0.327

The 95% confidence interval for the population proportion p is : (0.173 , 0.327)

At 90%

Z_/2 = Z _0.05 = 1.645

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

= 1.645 * (((0.250 * 0.750) / 120)

= 0.065

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.250 - 0.065< p < 0.250 + 0.065

0.185 < p < 0.315

The 95% confidence interval for the population proportion p is : (0.185 , 0.315)

c)

No, since the confidence interval contains the value 0.310.

d)

No, since the confidence interval contains the value 0.310.