Estimation in Sampling

GTECH 201Lecture 15

Conceptual Setting

How do we come to conclusions from empirical evidence? Isn’t common sense enough? Why?

Systematic methods for drawing conclusions from data Statistical inference

Inductive versus Deductive Reasoning

Drawing Conclusions Statistical inference

Based on the laws of probability What would happen if?

You ran your experiment hundreds of times You repeated your survey over and over again

Statistic and Parameter The proportion of the population who are

<disabled> usually denoted by: p In a SRS of 1000 people, the proportion of

the people who are <disabled> usually denoted by: (p -hat)

p̂

Estimating with Confidence

Say you are conducting an opinion poll… SRS of 1000 adult television viewers You ask these folks if they trust Walter

Cronkite when he delivers the nightly news Out of 1000, 570 say, they trust him 57% of the people trust Walter is 0.57 If you collect another set of 1000 television

viewers, what will the rating be?

p̂

Confidence Statement We need to add a confidence statement We need to say something about the

margin of error Confidence statements are based on

the distribution of the values of the sample proportion that would occur if many independent SRS were taken from the same population

The sampling distribution of the statistic

p̂

p̂

Terminology Review Sample Population Statistic

a numerical characteristic associated with a sample

Parameter A numerical characteristic associated with the

population Sampling error

The need for interval estimation

Point Estimation

Point estimation of a parameter is the value of a statistic that is used to estimate the parameter Compute statistic (e.g., mean) Use it to estimate corresponding

population parameter Point Estimators of Population Parameters

(see next slide)

Point Estimators for Population Parameters

x

ni

i

i

n

x

1

ni

i

i

n

xx

1

2

1

)( s

n

x

Tn

XNXN i)(

p

Population Sample CalculatingParameter statistic formula

Interval Estimation Sample point estimators are usually not

absolutely precise How close or how distant is the calculated

sample statistic from the population parameter

We can say that the sample statistic is within a certain range or interval of the population parameter.

The determination of this range is the basis for interval estimation

Interval Estimation (2)

A confidence interval (CI) represents the level of precision associated with a population estimate

Width of the interval is determined by Sample size, variability of the population, and the probability level or the level of confidence

selected

Sampling Distributionof the Mean

The distribution of all possible sample means for a sample of a given size

Use the mean of a sample to estimate and draw conclusions about the mean of that entire population

So we have samples of a particular size We need formulas to determine the mean and the

standard deviation of all possible sample means for samples of a given size from a population

Sample and Population Mean

For samples of size n, mean of the variable Is equal to the mean of the variable

under consideration Mean of all possible sample means is

equal to the population mean x

X

Sample Standard Deviation

For samples of size n, the standard deviation of the variable Is equal to the standard deviation of the

variable under consideration, divided by the square root of the sample size

For each sample size, the standard deviation of all possible sample means equals the population standard deviation divided by the square root of the sample size

nx

X

Central Limit Theorem Suppose all possible random samples of

size n are drawn from an infinitely large, normally distributed population having a mean and a standard deviation

The frequency distribution of these sample means will have: A mean of (the population mean) A normal distribution around this population

mean A standard deviation of

nx

Sampling Error Standard Error of the mean (SEM) is a basic

measure for the amount of sampling error

SEM indicates how much a typical sample mean is likely to differ from a true population mean

Sample size, and population standard deviation affect the sampling error

nx

Sampling Error (2) The larger the sample size, the smaller

the amount of sampling error The larger the standard deviation, the

greater the amount of sampling errorLarge

LargeSmall

Small

Sample size (n)Standard deviatio

n of populatio

n ()

Finite Population Correction Factor

The frequency distribution of the sample means is approximately normal if the sample size is large

N < 30 (small sample); N > 30 (large sample) If you have a finite population, then you need to

introduce a correction, i.e., the fpc rule/factor in the estimation process

where fpc = finite population correction; n = sample size; N = population size

1

N

nNfpc

Standard Error of the Mean for Finite

Populations

When including the fpc should be:

In general, you include the fpc in the population estimates only when the ratio of sample size to population size exceeds 5 % orwhen n / N > 0.05

)( fpcn

x

Constructing Confidence Intervals

A random sample of 50 commuters reveals that their average journey-to-work distance was 9.6 miles

A recent study has determined that the std. deviation of journey-to-work distance is approximately 3 miles

What is the CI around this sample mean of 9.6 that guarantees with 90 % certainty that the true population mean is enclosed within that interval?

Confidence Intervalfor the Mean

Z value associated with a 90 % confidence level (Z =1.65)

The sample mean is the best estimate of the true population mean

CI = 9.6 +1.65 (3/ ) = 10.30 miles 9.6 - 1.65 (3/ ) = 8.90 miles

xzx

50

3

6.9

n

x

5050

Confidence Interval We say that the sample statistic is within a certain

range or interval of the population parameter e.g., in our sample, 57% of the viewers thought Walter

Cronkite is trustworthy In the general population, between 54% and 60%

of viewers think that Walter Cronkite is trustworthy

Or, in our sample, the average commuting distance was 9.6 miles

In the population, we calculated that the average commute is likely to be somewhere between 8.9 miles and 10.3 miles

Confidence Level Gives you an understanding of how reliable your

previous statement regarding the confidence interval is

The probability that the interval actually includes the population parameter

For example, the confidence level refers to the probability that the interval (8.9 miles to 10.3 miles) actually encompasses the TRUE population mean (90%, 95%, 99.7%)

Confidence Level probability is 1 -

Significance Level

(alpha) The probability that the interval that

surrounds the sample statistic DOES NOT include the population parameter

E.g., the probability that the average commuting distance does not fall between 8.9 miles and 10.3 miles

= 0.10 (90%); 0.05 (95%); 0.01 (99.7%) Confidence Interval width -- increases

Sampling Error

Total sampling error = Probability that the sample statistic will

fall into either tail of the distribution is:

/2 If you want 99.7% confidence (i.e., low

error), then you have to settle for giving a less precise estimate (the CI is wider)

If the Standard Deviationis Unknown

If we don’t know the population mean, its likely we don’t know the standard deviation

What you are likely to have is the variance and standard deviation of your sample

Also, you have a small population, so you have to use the finite population correction factor that was discussed earlier

Once you have the formula for standard error, then you can proceed as before to determine the confidence interval

Standard Error

)( fpcn

x

1

N

nNfpc

N

nN

n

sx

2

n N2s

xzxCI

Student’s T Distribution

William Gosset (1876-1937) Published his contributions to

statistical theory under a pseudonym Student’s t distribution is used in

performing inferences for a population mean, when,

The population being sampled is approximately normally distributed

The population standard deviation is unknown

And the sample size is small (n < 30)

Characteristics of the t - Distribution

A t curve is symmetric, bell shaped Exact shape of distribution varies with

sample size When n nears 30, the value of t approaches

the standard normal Z value A particular distribution is identified by

defining its degrees of freedom (df) For a t distribution, df = (n -1)

n

sx

t

Properties of t Curves The total area under a t curve = 1 A t curve extends indefinitely in both

directions, approaching, but never touching the horizontal axis

A t-curve is symmetrical about 0 As the degrees of freedom become larger,

t curves look increasingly like the standard normal curve

We need to use a t-table and look for values of t, instead of Z to determine the confidence interval

Calculating various CIs Sampling

SRS, systematic, or stratified Parameters

Mean, total, or proportion Six situations

Consider whether to use fpc when n/N > 0.05

Consider whether to use Z or t when n < 30

If Random or Systematic Sample

Estimate of Population Mean Best estimate is ?

Estimate of sampling error Standard error of the mean (inc. fpc)

N

nN

n

sx

2

xzxCI

If Stratified Sample

Estimate of population mean Still equal to sample mean but…

Std. Error of the mean (inc. fpc)

mi

iii XN

NX

1

1

Where m=number of strata; i= refers to a particular stratum

i

iimi

i i

iix N

nN

n

sN

N 1

22

2

1

Minimum Sample Size

Before going out to the field, you want to know how big the sample ought to be for your research problem

Sample must be large enough to achieve precision and CI width that you desire

Formulas to determine the three basic population parameters with random sampling

Sample Size Selection - Mean

Your goal is to determine the minimum sample size You want to situate the estimated

population mean, in a specified CI

xzxCI

E = amount of error you are willing to tolerate

E

Zn

nZ

ZE x

2

Example 1

We are looking at Neighborhood X 3,500 households Sample size = 25 households Sample mean = 2.73 Sample variance = 2.6 CI = 90% Find the mean number of people per

household

Example 2 Sample of 30 households Sample standard deviation is 1.25 What sample size is needed to

estimate the mean number of persons per household in neighborhood X and be 90% confident that your estimate

will be within 0.3 persons of the true population mean?