9

Question

5
%
confidence interval for
Lower bound of the interval is
0
.
1
5
6
Upper bound of the interval is
0
.
3
2
4
9
9
%
confidence interval for
Lower bound of the interval is
0
.
1
3
0
Upper bound of the interval is
0
.
3
5
Which interval is wider
(
9
5
%
or
9
9
%
)
?

[
Select
]
Will it always be the case?

Answer 1

To determine which confidence interval is wider, we can compare the widths of the two intervals given:

1. For the 95% confidence interval:

   • Lower bound: \( 0.156 \)
   • Upper bound: \( 0.3249 \)
   • Width: \( 0.3249 - 0.156 = 0.1689 \)

2. For the 99% confidence interval:

   • Lower bound: \( 0.130 \)
   • Upper bound: \( 0.35 \)
   • Width: \( 0.35 - 0.130 = 0.22 \)

Now comparing the widths:

• 95% confidence interval width = \( 0.1689 \)
• 99% confidence interval width = \( 0.22 \)

Since \( 0.22 > 0.1689 \), the 99% confidence interval is wider than the 95% confidence interval.

Answer to the second question:

Generally, yes, higher confidence levels (e.g., increasing from 95% to 99%) will result in wider confidence intervals. This is because higher confidence levels require a larger margin of error to account for variability, thereby leading to a wider interval to ensure the estimated range contains the population parameter with the specified level of confidence.