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
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
Option A)
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