Question

A survey of 300 union members in New York State reveals that 112 favor the Republican candidate for governor. Construct the 90% confidence interval for the true population proportion of all New York State union members who favor the Republican candidate. Group of answer choices 0.308 < p < 0.438 0.301 < p < 0.445 0.316 < p < 0.430 0.327 < p < 0.419

Answer 1

To construct the 90% confidence interval for the true population proportion, we can use the formula: p' - EBP < p < p' + EBP. The sample proportion is 0.3733 and the margin of error is calculated to be 0.0643. Therefore, the confidence interval is 0.309 < p < 0.437.

Step-by-step explanation:

To construct a confidence interval for the true population proportion, we can use the formula:

p' - EBP < p < p' + EBP

Where p' is the sample proportion, EBP is the margin of error, and p is the true population proportion.

In this case, the sample proportion is 112/300 = 0.3733. The margin of error can be calculated using the formula:

EBP = Z * sqrt((p' * (1 - p')) / n)

Assuming a 90% confidence level, the Z value is 1.645. The sample size is 300. Calculating the margin of error:

EBP = 1.645 * sqrt((0.3733 * (1 - 0.3733)) / 300) = 0.0643

The confidence interval is:

0.309 < p < 0.437

Answer 2

Complete Question

A survey of 300 union members in New York State reveals that 112 favor the Republican candidate for governor. Construct the 98% confidence interval for the true population proportion of all New York State union members who favor the Republican candidate. Group of answer choices

A 0.308 < p < 0.438

B 0.301 < p < 0.445

C 0.316 < p < 0.430

D 0.327 < p < 0.419

Answer:

The correction option is A

Step-by-step explanation:

From the question we are told that

The sample size is n = 300

Th number that are in favor is k = 112

Generally the sample proportion is mathematically represented as

`\^ p = (k)/(n)`

=>
`\^ p = (112)/(300)`

=>
`\^ p = 0.3733`

From the question we are told the confidence level is 98% , hence the level of significance is

`\alpha = (100 - 98 ) \%`

=>
`\alpha = 0.02`

Generally from the normal distribution table the critical value of
`(\alpha )/(2)` is

`Z_{(\alpha )/(2) } =  2.33`

Generally the margin of error is mathematically represented as

`E =  Z_{(\alpha )/(2) } * \sqrt{(\^ p (1- \^ p))/(n) }`

=>
`E = 2.33 * \sqrt{(0.3733  (1- 0.3733))/(300) }`

=>
`E =  0.06508`

Generally 95% confidence interval is mathematically represented as

`\^ p -E <  p <  \^ p +E`

=>
`0.3733 -0.06508 <  p < 0.3733 + 0.06508`

=>
`0.308 <  p < 0.4038`