Chapter 12 - Inference About A Population

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Transcript Chapter 12 - Inference About A Population

Chapter 12

Inference About A Population

12.1

Inference About A Population…

Population Sample Inference Statistic Parameter •

Identify

the parameter to be estimated or tested.

•

Specify

the parameter’s estimator distribution.

and its sampling •

Derive the interval estimator and test statistic

12.2

Inference About A Population…

Population Sample Inference Statistic Parameter We will develop techniques to estimate and test three population parameters: Population Mean Population Variance Population Proportion

12.3

Inference With Variance Unknown…

Previously, we looked at estimating and testing the population mean when the population standard deviation ( ) was known or given: But how often do we know the actual population variance? Instead, we use the

Student t-statistic

, given by: 12.4

Inference With Variance Unknown…

When is unknown, we use its point estimator

and the z-statistic is replaced by the the t-statistic, where the number of “degrees of freedom” , is n–1.

12.5

Testing when is unknown…

When the population standard deviation is unknown and the population is normal, the test statistic for testing hypotheses about is: which is Student

distributed with = n–1 degrees of freedom. The confidence interval estimator of is given by: 12.6

Example 12.1…

Will new workers achieve 90% of the level of experienced workers within one week of being hired and trained?

Experienced workers can process 500 packages/hour, thus if our conjecture is correct, we expect new workers to be able to process .90(500) = 450 packages per hour.

Given the data , is this the case?

12.7

Example 12.1…

IDENTIFY

Our objective is to

describe

the population of the numbers of packages processed in 1 hour by new workers, that is we want to know whether the new workers’ productivity is more than 90% of that of experienced workers. Thus we have: H 1 : > 450 Therefore we set our usual null hypothesis to: H 0 : = 450 12.8

Example 12.1…

Our test statistic is:

COMPUTE

With n=50 data points, we have n–1=49 degrees of freedom. Our hypothesis under question is: H 1 : > 450 Our rejection region becomes: Thus we will reject the null hypothesis in favor of the alternative if our calculated test static falls in this region.

12.9

Example 12.1…

COMPUTE

From the data, we calculate = 460.38, s =38.83 and thus: Since we reject H 0 in favor of H 1 , that is, there is sufficient evidence to conclude that the new workers are producing at more than 90% of the average of experienced workers.

12.10

Example 12.1…

COMPUTE

Alternatively, we can use

t-test:Mean

12.11

Example 12.1…

COMPUTE

In addition to looking at the computed t-statistic and the critical value of t (one tail), we could look at the p-value (0.0323) and see that it is “small” (~3%), so again, we reject the null hypothesis in favor of the alternative… 12.12

Example 12.2…

IDENTIFY

Can we estimate the return on investment for companies that won quality awards?

We have are given a random sample of n = 83 such companies. We want to construct a 95% confidence interval for the mean return, i.e. what is: ??

12.13

Example 12.2…

COMPUTE

12.14

Example 12.2…

INTERPRET

We are 95% confident that the population mean, , i.e. the mean return of all publicly traded companies that win quality awards, lies between 13.20% and 16.84% Tools > Data Analysis Plus >

t-Estimate: Mean

is an alternative to the manual calculation… 12.15

Check Requisite Conditions…

The Student t distribution is population is “not

extremely robust

, which means that if the population is nonnormal, the results of the t-test and confidence interval estimate are still valid provided that the nonnormal”.

To check this requirement, this case.

draw a histogram

of the data and see how “bell shaped” the resulting figure is. If a histogram is extremely skewed (say in the case of an exponential distribution), that could be considered “extremely nonnormal” and hence t-statistics would be not be valid in 12.16

Estimating Totals of Finite Populations…

Large populations

are defined as “populations that are at least 20 times the sample size” We can use the confidence interval estimator of a mean to produce a confidence interval estimator of the

population total

: Where N is the size of the finite population.

12.17

Estimating Totals of Finite Populations…

For example, a sample of 500 households (in a city of 1 million households) reveals a 95% confidence interval estimate that the

household mean

spent on Halloween candy lies between $20 & $30.

We can estimate the multiplying these lower and upper confidence limits by the total population:

total

amount spent in the city by Thus we estimate that the

total

amount spent on Halloween in the city lies between $20 million and $30 million.

12.18

Identifying Factors…

12.19

Inference About Population Variance…

If we are interested in drawing inferences about a

population’s variability

, the parameter we need to investigate is the population variance: The sample variance (

s 2

) is an unbiased, consistent and efficient point estimator for . Moreover, the statistic, , has a chi-squared distribution, with n–1 degrees of freedom.

12.20

Testing & Estimating Population Variance

The test statistic used to test hypotheses about is: (which is chi-squared with = n–1 degrees of freedom).

12.21

Testing & Estimating Population Variance

Combining this statistic: With the probability statement: Yields the

confidence interval estimator for

upper confidence limit 12.22

Example 12.3…

IDENTIFY

Consider a container filling machine. Management wants a machine to fill 1 liter (1,000 cc’s) so that that variance of the fills is less than 1 cc significance level?

2 . A random sample of n=25 1 liter fills were taken. Does the machine perform as it should at the 5% We want to show that: Variance is less than 1 cc 2 H 1 : < 1 (so our null hypothesis becomes: H 0 : = 1). We will use this test statistic: 12.23

Example 12.3…

COMPUTE

Since our alternative hypothesis is phrased as: H 1 : < 1 We will reject H 0 in favor of H 1 this rejection region: if our test statistic falls into We computer the sample variance to be: s 2 =.8088

And thus our test statistic takes on this value… 12.24

Example 12.3…

Since:

INTERPRET

There is not enough evidence to infer that the claim is true.

Excel output can also be used for this test… compare 12.25

Example 12.4…

As we saw, we cannot reject the null hypothesis in favor of the alternative. That is, there is not enough evidence to infer that the claim is true.

Note:

the result does not say that the variance is greater than 1, rather it merely states that we are

unable

to show that the

variance is less than 1

We could estimate (at 99% confidence say) the variance of the fills… 12.26

Example 12.4…

COMPUTE

In order to create a confidence interval estimate of the variance, we need these formulae: lower confidence limit upper confidence limit we know (n–1)s 2 = 19.41 from our previous calculation, and we have from Table 5 in Appendix B: 12.27

Example 12.4…

Thus the 99% confidence interval estimate is:

COMPUTE

That is, the variance of fills lies between .426 and 1.963 cc 2 .

12.28

Identifying Factors…

12.29

Inference: Population Proportion…

When data are nominal, we count the number of occurrences of each value and calculate proportions. Thus, the parameter of interest in describing a population of nominal data is the population proportion

This parameter was based on the binomial experiment.

Recall the use of this statistic: where p-hat ( ) is the sample proportion:

sample size of

items.

12.30

Inference: Population Proportion…

When

and

n(1–p)

are both greater than 5, the sampling distribution of is approximately normal with mean: standard deviation: Hence: 12.31

Inference: Population Proportion…

Test statistic for

: The confidence interval estimator for

is given by: (both of which require that

>5 and

n(1–p)

12.32

Example 12.5…

IDENTIFY

At an exit poll , voters are asked by a certain network if they voted Democrat (code=1) or Republican (code=2). Based on their small sample, can the network conclude that the Republican candidate will win the vote?

That is: H 1 :

> .50

And hence our null hypothesis becomes: H 0 :

= .50 12.33

Example 12.5…

Since our research hypothesis is: H 1 :

> .50

our rejection region becomes:

COMPUTE

Looking at the data, we count 407 (of 765) votes for code=2. Hence, we calculate our test statistic as follows… 12.34

Example 12.5…

Since:

INTERPRET

…we reject H 0 in favor of H 1 , that is, there is enough evidence to believe that the Republicans win the vote.

Likewise from Excel: compare these… …or look at p-value 12.35

Selecting the Sample Size…

The confidence interval estimator for a population proportion is: Thus the (half)

width

of the interval is: Solving for

12.36

Selecting the Sample Size…

For example, we want to know how many customers to survey in order to estimate the proportion of customers who prefer our brand to within .03 (with 95% confidence).

I.e. our confidence interval after surveying will be ± .03, that means W=.03

Uh Oh. Since we haven’t taken a sample yet, we don’t have this sample proportion… Substituting into the equation… 12.37

Selecting the Sample Size…

Two methods – in each case we choose a value for then solve the equation for n.

Method 1 : no knowledge of even a rough value of . This is a ‘worst case scenario’ so we substitute = .50

Method 2 : we have some idea about the value of . This is a better scenario and we substitute in our estimated value.

12.38

Selecting the Sample Size…

Method 1 : no knowledge of value of , use 50%: Method 2 : some idea about a possible value, say 20%: Thus, we can sample fewer people if we already have a reasonable estimate of the population proportion before starting.

12.39

Estimating Totals for Large Populations…

In much the same way as we saw earlier, when a population is

large

and

finite

of the population ( we can estimate the total number of successes in the population by taking the product of the size

) and the confidence interval estimator: The Nielsen Ratings (used to measure TV audiences) uses this technique. Results from a small sample audience (say 2,000 viewers) is extrapolated to the total number of TV sets (say 100 million)… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

12.40

Nielsen Ratings Example…

COMPUTE

Problem: describe the population of television shows watched by viewers across the country (population), by examining the results from 2,000 viewers (sample).

We take these values and multiply them by are watching the “Tonight Show”.

12.41

Identifying Factors…

Factors that identify the z-test and interval estimator of

12.42

Flowchart of Techniques…

Describe a Population Central Location

t test & estimator of u.

Data Type?

Interval

Type of descriptive measurement?

Nominal

z test & estimator of p

Variability

X 2 test & estimator of s 2

12.43

Chapter 12 - Inference About A Population

Transcript Chapter 12 - Inference About A Population

Chapter 12

Inference About A Population

Inference About A Population…

Inference About A Population…

Inference With Variance Unknown…

Inference With Variance Unknown…

Testing when is unknown…

Example 12.1…

Example 12.1…

Example 12.1…

Example 12.1…

Example 12.1…

Example 12.1…

Example 12.2…

Example 12.2…

Example 12.2…

Check Requisite Conditions…

Estimating Totals of Finite Populations…

Estimating Totals of Finite Populations…

Identifying Factors…

Inference About Population Variance…

Testing & Estimating Population Variance

Testing & Estimating Population Variance

Example 12.3…

Example 12.3…

Example 12.3…

Example 12.4…

Example 12.4…

Example 12.4…

Identifying Factors…

Inference: Population Proportion…

Inference: Population Proportion…

Inference: Population Proportion…

Example 12.5…

Example 12.5…

Example 12.5…

Selecting the Sample Size…

Selecting the Sample Size…

Selecting the Sample Size…

Selecting the Sample Size…

Estimating Totals for Large Populations…

Nielsen Ratings Example…

Identifying Factors…

Flowchart of Techniques…

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