Estimation
Confidence Intervals
Learning Objectives:
- Define confidence intervals and confidence levels.
- Construct a confidence interval for specific parameters.
- Estimate the sample size for a required confidence interval.
A Confidence Interval is a specific interval estimate of a parameter determined by using data obtained from a sample and the specific confidence level of the estimate.
A Confidence Level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.
Confidence Interval:
Part A: When $\sigma$ is known:Part B: When $\sigma$ is unknown:
where $t_{\frac{\alpha}{2}}$ corresponds to degree of freedom = n-1
Ruby00269, based on chart by Shishirdasika,
CC BY-SA 3.0, via Wikimedia Commons
Solved Example: 9950-01
Given a Mean=100, SD=20 and N =100. Which one of the following will be the correct statement for indicating the possibility of population mean at 0.01 level of confidence?
A. Population mean falls between 93 to 107
B. Population mean falls between 96.08 to 103.9
C. Population mean falls between 97.2 to 102
D. Population mean falls between 94.8 to 105.2
0.01 level of confidence means correponding to 99% area, for which Z = 2.58 (standard value from the table Page 75)
Confidence interval is given by:
\begin{align*}
x &= \bar{x} \pm Z \dfrac{\sigma}{\sqrt{N}}\\
&= 100 \pm 2.58 \dfrac{20}{\sqrt{100}}\\
&= 100 \pm 2.58 \times 2\\
&= 100 \pm 5.16
\end{align*}
Hence the lower limit = 100- 5.16 = 94.84
and the upper limit = 100 + 5.16 = 105.16
Correct Answer: D
Solved Example: 9950-02
A confidence interval increases in width as the:
A. Sample size is increased
B. Degrees of freedom is increased
C. Level of confidence increases
D. Sample standard deviation is decreased
As the confidence level increase, more and more area is included in the acceptance region. The lower limit and the upper limit spread farther apart and as a result, confidence interval width increases.
Correct Answer: C