QBA241 Handbook Topic 8 Estimation and CI

Estimation and CI

Topic 8 QBA241 2015

8. Estimation and Confidence Interval

Recall that statistical methods are divided into descriptive and inferential statistics. This topic of Estimation and Confidence Interval falls under the inferential statistics category.

Within this topic, there will be two parts: 1) Estimation and confidence interval and 2) Sample size determination.

8.1 Confidence Interval (also written as ‘CI’) for Population Mean

The population parameter such as the population mean (µ) is usually unknown. Estimating an unknown population parameter from a sample is known as a point estimate. A point estimate is a single number. Thus, the sample mean (???) is a single point estimate for the population mean (µ). The confidence interval provides a range (an interval) of estimates for the unknown population parameter. It requires the value of the lower and the upper interval of the estimates. The lower and upper interval of the estimates is known as the Lower Confidence Limit (LCL) and Upper Confidence Limit (UCL) respectively. The distance between the LCL and UCL is known the width of the confidence interval.

In order to compute the each value of the LCL and UCL, the point estimate such as the sample mean (???) is required as the basis of estimation. In addition to that, the spread of the distribution (standard deviation) is necessary to determine the width in the interval.

Estimation and Confidence Interval 63 HH

Estimation and CI

Topic 8 QBA241 2015

The CI gives information about closeness to the unknown population parameter. The CI must be constructed with a given level of confidence such as 90%, 95% or 99% (never 100% confident). Typically, if the level of CI is 90% then the probability of 100% - 90% is equal to the level of significance (?). Alternatively, it can also be written as such: (1 – ?) = CI so (1 – 0.10) = 0.90. Take note that, the level of significance (?) is then divided into two equal tails. Refer to the illustrations below.

8.1.1 Confidence Interval for a Population Mean (? known)

The information required to construct a confidence interval for a population mean includes (assuming that the population is normally distributed):

? ? ? ?

Point estimate: sample mean (???)

Level of confidence (either 90%, 95% or 99%) Population standard deviation (?) Sample size (n)

Confidence Interval for a Population Mean (? known)

90% Confidence Interval

Z

?0

Z

?±????/??(??=??

where,

? µ = population mean ? ???

= sample mean

?

α

= level of significance

? ? = population standard deviation ? n = sample size

? Z score of standard normal distribution

??√

Estimation and Confidence Interval 64 HH

Estimation and CI

Topic 8 QBA241 2015

Estimation and Confidence Interval

65

HH

Estimation and CI

Topic 8 QBA241 2015

Interpreting the Confidence Interval Estimate

Refer to the Example 8.1.1. Given the above, with 95% confidence, the population mean lightbulbs will be between 1033.71 and 1089.49 hours. This notation means that if we repeatedly draw samples of 100 from this population of lightbulbs, 95% of the values of ??? will be such that µ would

??27.8913 ?????????) and (???+27.8913 ?????????). 10% of the values of ??? lie somewhere between (??

will produce intervals that would not include µ.

8.1.2 Confidence Interval for a Population Mean (? unknown)

In the situation when the population standard deviation (?) is unknown, the sample standard deviation (s) becomes the point estimate. Hence, the information required to construct a confidence interval for a population mean (? unknown) includes the following. The population is assumed to be normally distributed:

?

?

?

? Point estimate: sample mean (???) Level of confidence (either 90%, 95% or 99%) Sample standard deviation (s) Sample size (n)

Confidence Interval for a Population Mean (? unknown)

?±????(??=????√

where,

? µ = population mean

? ??? = sample mean

? α = level of significance

? s = sample standard deviation

? n = sample size

? Student t distribution based on α/2 and (n – 1)

Estimation and Confidence Interval 66 HH

Estimation and CI

Topic 8 QBA241 2015

Estimation and Confidence Interval 67 HH