Question

A 95% confidence interval for the proportion of college students who have texted during a class was 0.75 to 0.95. Which of the following is the 90% confidence interval from the same sample?

A

0.766 to 0.934

B

0.777 to 0.923

C

0.731 to 0.969

D

0.05 to 0.25

Answer #1

Solution :

Given that,

Lower confidence interval = 0.75

Upper confidence interval = 0.95

Point estimate = = (Lower confidence interval + Upper confidence interval ) / 2

= (0.75 + 0.95) / 2

= 1.7 / 2

= 0.85

1 - = 1 - 0.85 = 0.15

Margin of error = E = Upper confidence interval - = 0.95 - 0.85 = 0.10

sample size = n = (Z_{
/ 2} / E )^{2} *
* (1 -
)

= (1.96 / 0.10)^{2} * 0.85 * 0.15

n = 49

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 * (((0.85 * 0.15) / 49)

= 0.084

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.85 - 0.084 < p < 0.85 + 0.084

0.766 < p < 0.934

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