1

Session 7

Standard errors, Estimation

and Confidence Intervals

2

By the end of this session, you will be able to explain what is meant by an estimate of a

population parameter, and its standard error

explain the meaning of a confidence interval

calculate a confidence interval for the population mean using sample data, and state the assumptions underlying the above calculation

Learning Objectives

3

Inference is about drawing conclusions concerning population characteristics using information gathered from the sample

It is assumed that the sample is representative of the population

A further assumption is that the sample has been drawn as a simple random sample from an infinite population

Reminder: What is inference?

4

Population Sample

Mean Variance 2 s2

Std. deviation s

x

Population characteristics (parameters) are denotedby greek letters, sample values by latin letters

Sample characteristics are measurable and form estimates of the population values.

Estimation

5

What is the mean number of personsper household in Mukono district?

Data from 80 households surveyed in thisdistrict gave a mean household size of 5.6with a standard deviation 3.30.

Hence our best estimate of the mean householdsize in Mukono district is therefore 5.6.

What results are likely if we sampled again with a different set of households?

Example of statistical inference

6

Example using Stata

Open Stata file UNHS_hh&poverty.dta

Numeric code is 109 for Mukono

7

Use summarize dialogueType db summarize or use menu

Statistics Summaries, tables Summaries Summary Statistics

Variable hhsize

Then use by/if/in tab

dist ==109 is condition

8

Results

Summaries for Mukono only

Summaries for whole sample

9

Suppose 10 University students were given astandard meal and the time taken to consumethe meal was recorded for each.

Suppose the 10 values gave:

mean = 11.24, with std.dev.= 0.864

Let’s assume this exercise was repeated 50 timeswith different samples of students

A histogram of the resulting 500 obs. appears below, followed by a histogram of the 50 means from each sample

The distribution of means

10

The data appear to follow a normal distribution

Histogram of raw data

11

The distn of the sample means is called its Sampling Distribution

Notice that the variability of the above distn is smaller than the variability of the raw data

Histogram of 50 sample means

12

The estimate of the mean household size inMukono district was 5.6.

Is this sufficient for reporting purposes, given that this answer is based on one particular sample?

What we have is an estimate based on a sample of size 80. But how good is this estimate?

We need a measure of the precision, i.e. variability, of this estimate…

Back to estimation…

13

The accuracy of the sample mean asan estimate of depends on:

(i) the sample size (n)

since the more data we collect, the morewe know about the population, and the

(ii) inherent variability in the data 2

These two quantities must enter the measureof precision of any estimate of a population parameter. We aim for high precision, i.e.low standard error!

x

Sampling Variability

14

Precision of as estimate of is given by:

the standard error of the mean.

Also written as s.e.m., or sometimes s.e.

It is estimated using the sample data: s/n

For example on household size,

s.e.=3.298/80 = 3.298/8.944 = 0.369

Standard error of the mean

s.e. x n

x

15

Instead of using a point estimate, it is usually more informative to summarise using an interval which is likely (i.e. with 95% confidence) to contain .

This is called an interval estimate or a Confidence Interval (C.I.)

For example, we could report that the mean household size of HHs in Mukono district is 5.6 with 95% confidence interval (4.87, 6.33), i.e. there is a 95% chance that the interval (4.87,6.33) includes the true value .

Confidence Interval for

16

Analysis using Stata

Type db ci or use menu

Use the by/if/in tab as before

17

Results

For whole sample

Just for Mukono

18

The 95% confidence limits for (lower and upper)are calculated as:

and

2½%2½%

0–t t

where tn-1 is the 5% level for

the t-distribution with (n-1) degrees of freedom.Statistical tables and statistical software give t-values.

1 ( )nx t s n 1 ( )nx t s n

Finding the Confidence Interval

19

2½%2½%

0–t t

P 10 5 2 = 1 6.31 12.7 31.82 2.92 4.30 6.963 2.35 3.18 4.544 2.13 2.78 3.755 2.02 2.57 3.36

6 1.94 2.45 3.147 1.89 2.36 3.008 1.86 2.31 2.909 1.83 2.26 2.8210 1.81 2.23 2.76

20 1.72 2.09 2.5330 1.70 2.04 2.4640 1.68 2.02 2.4260 1.67 2.00 2.39

1.64 1.96 2.33

1 ( )nx t s n

t-values for finding 95% C.I.

20

10

11

12

13

0 5 10 15 20 25 30 35 40 45 50

If we sampled repeatedly and found a95% C.I. each time, only 95% of themwould include the true , i.e. there is a 95% chance that a single interval would include .

Correct interpretation of C.I.s

21

For rural households (n=40) in Mukono, wefind mean=6.43, std.dev.=3.54 for the numberof persons per household.

Hence a 95% confidence interval for the true mean number of persons per household:

6.43 t39 (s/n) = 6.43 2.02(3.54/40)

= 6.43 1.13

= (5.30, 7.56)

Can you interpret this interval? Write down your answer. We will then discuss.

An example (persons per HH)

22

Analysis in StataPress Page Up to retrieve the last commandThen add “& rurban == 0” to the condition

Or use the menus and change the dialogue

23

The above computation of a confidence interval assumes that the data have a normal distribution.

More exactly, it requires the sampling distribution of the mean to have a normal distribution.

What happens if data are not normal?

Not a serious problem if sample size is large because of the Central Limit Theorem, i.e. that the sampling distribution of the mean has a normal distribution, for large sample sizes.

Underlying assumptions

24

So even when data are not normal, the formula for a 95% confidence interval will give an interval whose “confidence” is still high - approximately 95%.

It is better to attach some measure of uncertainty than worry about the exact confidence level.

Assumptions - continued