1

Day 3QMIM

Confidence Interval for a Population mean of Large and Small Samples

by Binam Ghimire

2

To be able to calculate best estimates of the mean and standard deviation of a population

To be able to calculate confidence intervals for a population mean for large and small samples

Objective

3

Why Sampling? (1)

Alternate to sampling is to test the entire population.

The advantage of testing the entire population is the accuracy

The disadvantages are 1) Expensive, 2) Time consuming 3) Destructive and 4) not possible or total population unknown

4

Why Sampling? (2)

Car Crash Test

Water Resistant Test – The deep dive watch from Rolex

Examples

5

Point Estimates

Population Sample

Mean m

Std. deviation s s

Size N n

Symbols

6

Estimate for the population parameter:Conditions

Sample should be part of population

Sample should represent the population

Sample should be random

Larger the better

7

Provides Range of Values Based on Observations from Sample

Gives Information about closeness to unknown population parameter

Stated in terms of Probability 90%, 95 %, 99%

Confidence Interval Estimations

8

Probability that the population parameter falls within certain range

Lower Range Higher Range

Confidence Interval Estimations

9

Point EstimateLower Confidence Limit

UpperConfidence Limit

Width of confidence interval

Confidence Interval Estimations

10

Confidence Interval Estimations

For largest possible sample, margin of error = 0. (This will happen when sample data = population data)

If so thenm =

But the is from sample so will not be exactly equal to m. In fact, it will be either lower or higher than the u [Never 100% (1 – a) ]

m = +/ - Error ... (1)

11

Confidence Interval Estimations:Central Limit TheoremWe may like to arrive close to m by finding means from multiple samples. If population is normally distributed then the “sampling distribution of the means” will also be normal. If the population is not normally distributed then whether “sampling distribution of the mean” will be normal or not depends on the size of the sample – Central Limit Theorem.

The spread of the sampling distribution also depends upon the sample size. Larger the sample size the narrower the spread (or smaller the standard deviation)

12

Source: Oakshott, L. (2006) Essential Quantitative Methods for Business, Management and Finance, 3 rd Ed., Palgrave, p. 226

Confidence Interval Estimations:Central Limit Theorem

13

m = +/ - Error ... (1)

Error depends on two factors: 1) standard deviation (s) and 2) sample size (n).

We call the error (standard deviation of the sampling distribution) Standard Error

So we may call the above equation as follows

m= +/ - Standard Error ... (2)

Standard Error =

Confidence Interval Estimations:Standard Error

14

Standard error of sample mean is the standard deviation of the distribution of sample means.

When the population σ is known:

When the population σ is unknown:

Confidence Interval Estimations:Standard Error

15

Confidence Interval Estimations:Deriving the Formula

When the number of sample is larger than 30 we can apply the normal distribution to calculate the limits of confidence interval (Because Z score table is for sample size > 30).

Replacing x by , and by Standard Error ( ) we get,

... (3)

16

Common Levels of confidence:Checking the Z table

For a 99% confidence Interval, Z = _____

For a 95% confidence Interval, Z = _____

For a 90% confidence Interval, Z = _____

17

Exercise

If a sample of 100 items is drawn from a population and the mean is found to be 200g with a standard deviation of 5g, find: (a)a 95% confidence interval estimate for

the population mean.(b)if a sampling error of only ± 0.5g is

allowed, calculate the size of sample

18

Small Samples – T Distribution

19

T-Distribution

20

Small Samples – T Distribution

Sample size < 30 = T-distribution Properties of t-distribution (Student’s t-

distribution) Symmetrical (bell shaped) Less peaked and fatter tails than a normal

distribution Defined by single parameter, degrees of

freedom (df), where df = n – 1 As df increase, t-distribution approaches

normal distribution

21

T table The number within the table are t-values not

probabilities Numbers in the first column are degree of

freedom (v) of the sample This is the freedom that you have in choosing

the values of the sample. If you were given the mean of the sample of 8 values, you would be free to choose 7 of the 8 but not the 8th one. Therefore there are 7 degrees of freedom

The number of degrees of freedom is therefore n-1

For a very large sample the t and Z distributions are same

For a 95% confidence interval you will choose 0.025 level

Properties of t-distribution (Student’s t-distribution) Symmetrical (bell shaped) Less peaked and fatter tails than a normal

distribution Defined by single parameter, degrees of

freedom (df), where df = n – 1 As df increase, t-distribution approaches

normal distribution

22

T-Distribution

The formula is same like Z, we just replace the Z by t

23

ExerciseSample mean is 494.6 and standard deviation is 23.03. Calculate the confidence interval estimates for a sample size of 10 at 95% confidence interval