QBM117 - Business Statistics

Estimating the population mean , when the population variance 2, is unknown

Estimating the population mean when the population variance 2 is unknown

However is not normally distributed.

ns

x

/

xs x

In reality, if we do not know the population mean , it is unlikely that we will know the population standard deviation .

Therefore we use the sample standard deviation, s, to estimate the population standard deviation and hence the standard error to estimate

W.S Gosset showed that has a particular

distribution called the student t distribution or simply the t distribution when the population from which the sample is drawn is normal.

ns

x

/

ns

xt

/

is called the t statistic

What if the population from which we are sampling is not normal?

The t distribution is said to be.robust. This means that the t distribution also provides an adequate approximate sampling distribution of the t statistic for moderately non-normal populations.

In actual practice, we should draw the histogram of any random variable that you are assuming is normal, to ensure that the assumption is not badly violated.

If the assumption is not satisfied at all, due to extreme skewness, we have two options:• transform the data (perhaps with logarithms) to bring about a

normal distribution, or

• use non parametric methods (studied in QBM217)

What do we know about the t distribution? It looks very much like the standard normal probability

density function, but with fatter tails and slightly more rounded peaks.

It is more widely dispersed than the normal probability density function.

The graph of the t probability density function changes for different sample sizes.

The t statistic has n - 1 degrees of freedom. The similarity between the t pdf and the standard normal

pdf increases rapidly, as the degrees of freedom for the t pdf increases.

The two distributions are virtually indistinguishable when the degrees of freedom exceed 30.

The values for are identical to the corresponding ,2/t 2/Z

Estimating the population mean when the population variance 2 is unknown

x

n

s

n

stx n 1.2/

The (1-α)100% confidence interval for µ is given by

where

is the sample mean

1.2/ nt is the value of t for the given level of confidence (S&S Table 4 in appendix)

is the standard deviation of the sample mean, known as the standard error

Example 1 – Exercise 8.14 p264

Here we want to estimate the population mean . The sample mean is the best estimator of . We have sampled from a normal population therefore,

will follow the t distribution.

x

ns

x

/

38.23.27

)75

8.7)(648.2(3.271,2/

n

stx n

Therefore the confidence interval is given by

68.29to92.24)(%99 CI

648.201.099.01

8.7753.27

70,005.074,005.0

tt

snx

Two confidence interval estimators of We now have two different interval estimators of the population mean. The basis for determining which interval estimator to use is quite simple.

If is known the confidence interval estimator of the population mean is

nzx

2/

If is unknown and the population is normally distributed, the confidence interval estimator of the population mean is

n

stx n 1,2/

When the degrees of freedom exceed 200, we approximate the required t statistic by the value.,2/t

Example 2

A foreman in a manufacturing plant wishes to estimate the average amount of time it takes a worker to assemble a certain device. He randomly selects 81 workers and discovers that they take an average of 29 minutes with a standard deviation of 4.5 minutes. Assuming the times are normally distributed, find a 90% confidence interval estimate for the average amount of time it takes the workers in this plant to assemble the device. What can you report to the foreman?

Example 1

A foreman in a manufacturing plant wishes to estimate the average amount of time it takes a worker to assemble a certain device. He randomly selects 81 workers and discovers that they take an average of 29 minutes with a standard deviation of 4.5 minutes. Assuming the times are normally distributed, find a 90% confidence interval estimate for the average amount of time it takes the workers in this plant to assemble the device. What can you report to the foreman?

n

stx n 1,2/

Since is unknown and the population is normally distributed, the confidence interval estimator of the population mean is

83.029

)81

5.4)(664.1(291,2/

n

stx n

Therefore the confidence interval is given by

664.11.09.01

815.429

80,05.1,2/

tt

nsX

n

We are 90% confident that the mean assembly time lies between 28.17 and 29.83 minutes.

Example 3

A random sample of 26 airline passengers at the local airport showed that the mean time spent waiting in line to check in at the ticket counter was 21 minutes with a standard deviation of 5 minutes.

Construct a 99% confidence interval for the mean time spent waiting in line by all passengers at this airport.

Assume the waiting times for all passengers are normally distributed.

Reading for next lecture

S&S Chapter 8 Sections 8.5 - 8.7

Exercises to be completed before next lecture

S&S 8.21 8.23