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Parameter Estimation• Parameter estimation: the process of using sample information to compute an interval that describes the range of values of a parameter such as the population mean o

Generalizing a Sample’s Findings to Its Population and Testing Hypotheses About Percents and

Means

Statistics Versus Parameters

• Statistics: values that are computed

from information provided by a sample

• Parameters: values that are computed from a complete census which are

considered to be precise and valid

measures of the population

• Parameters represent “what we wish

to know” about a population Statistics are used to estimate population

The Concepts of Inference and

Statistical Inference

• Inference: drawing a conclusion

based on some evidence

• Statistical inference: a set of

procedures in which the sample size and sample statistics are used to

make estimates of population

parameters

How to Calculate Sample Error

(Accuracy)

n

pq z

error =

sp

Where z = 1.96 (95%)

or 2.58 (99%)

Accuracy Levels for Different

Parameter Estimation

• Parameter estimation: the process of using sample information to compute

an interval that describes the range

of values of a parameter such as the population mean or population

percentage is likely to take on

Parameter Estimation

• Parameter estimation involves three

values:

1 Sample statistic (mean or percentage

generated from sample data)

2 Standard error (variance divided by

sample size; formula for standard error of the mean and another formula for standard error of the percentage)

3 Confidence interval (gives us a range

within which a sample statistic will fall

Parameter Estimation

• Statistics are generated from sample data and are used to estimate population

parameters.

• The sample statistic may be either a

percentage, i.e., 12% of the respondents stated they were “very likely” to patronize a new, upscale restaurant OR

• The sample statistic may be a mean, i.e., the average amount spent per month in

restaurants is $185.00

Parameter Estimation

• Standard error: while there are two

formulas, one for a percentage and the other for a mean, both formulas have a measure of variability divided

by sample size Given the sample

size, the more variability, the greater the standard error

Parameter Estimation

• The lower the standard error, the

more precisely our sample statistic

will represent the population

parameter Researchers have an

opportunity for predetermining

standard error when they calculate

the sample size required to

accurately estimate a parameter

Recall Chapter 13 on sample size

Standard Error of the Mean

Standard Error of the

Percentage

Parameter Estimation

• Confidence intervals: the degree of accuracy desired by the researcher and stipulated as a level of

confidence in the form of a

percentage

• Most commonly used level of

confidence: 95%; corresponding to

1.96 standard errors

Parameter Estimation

• What does this mean? It means that

we can say that if we did our study

over 100 times, we can determine a range within which the sample

statistic will fall 95 times out of 100

(95% level of confidence) This gives

us confidence that the real population value falls within this range

• Theoretical notion

• Take many, many, many samples

• Plot the p’s

• 95 % will fall in confidence interval

How do I interpret the confidence

interval?

2.5% 2.5%

95%

Parameter Estimation

• Five steps involved in computing

confidence intervals for a mean or percentage:

1 Determine the sample statistic

2 Determine the variability in the

sample for that statistic

Parameter Estimation

3 Identify the sample size

4 Decide on the level of confidence

5 Perform the computations to

determine the upper and lower

boundaries of the confidence

interval range

Parameter Estimation Using SPSS: Estimating a Percentage

• Run FREQUENCIES (on

RADPROG) and you find that 41.3% listen to “Rock” music

• So, set p=41.3 and then q=58.7,

• The answer is 36.5%-46.1%

• We are 95% confident that the true %

of the population that listens to

“Rock” falls between 36.5% and

How to Compute a Confidence

Interval for a Percent

n

pq z

is 41.3 percent, and we are 95 percent confident that the true population value is between 36.5

Parameter Estimation Using SPSS: Estimating a Mean

• SPSS will calculate a confidence

interval around a mean sample

an upscale restaurant spend in

restaurants per month (See p

Parameter Estimation Using SPSS: Estimating a Mean

• We must first use DATA, SELECT

CASES to select LIKELY=5

• Then we run ANALYZE, COMPARE MEANS, ONE SAMPLE T-TEST

• Note: You should only run this test

when you have interval or ratio data

Parameter Estimation Using SPSS: Estimating a Percentage

• Estimating a Percentage: SPSS will not calculate for a percentage You must run FREQUENCIES to get your sample statistic and n size Then use

• AN EXAMPLE: We want to estimate the percentage of the population that listens to “Rock” radio

Estimating a Population

Percentage with SPSS

• Suppose we wish to know how accurately the sample statistic estimates the percent listening to “Rock” music.

– Our “best estimate” of the population

percentage is 41.3% prefer “Rock” music stations (n=400) We run

FREQUENCIES to learn this.

– But how accurate is this estimate of the

true population percentage preferring

rock stations?

Estimating a Population Mean

with SPSS

• How do we interpret the results?

– My best estimate is that those “very

likely” to patronize an upscale restaurant

in the future, presently spend $281

dollars per month in a restaurant In

addition, I am 95% confident that the true population value falls between $267 and

$297 (95% confidence interval)

Therefore, Jeff Dean can be 95%

confident that the second criterion for the

Hypothesis Testing

• Hypothesis: an expectation of what the population parameter value is

• Hypothesis testing: a statistical

procedure used to “accept” or “reject”

the hypothesis based on sample

information

• Intuitive hypothesis testing: when

someone uses something he or she has observed to see if it agrees with or

refutes his or her belief about that topic

Hypothesis Testing

• Statistical hypothesis testing:

– Begin with a statement about what you believe exists in the population– Draw a random sample and

determine the sample statistic

– Compare the statistic to the

hypothesized parameter

Hypothesis Testing

• Statistical hypothesis testing:

– Decide whether the sample

supports the original hypothesis

– If the sample does not support the hypothesis, revise the hypothesis

to be consistent with the sample’s statistic

What is a Statistical Hypothesis?

• A hypothesis is what someone

expects (or hypothesizes) the

population percent or the average

to be

• If your hypothesis is correct, it will

fall in the confidence interval

• If your hypothesis is incorrect, it will fall outside the confidence interval

How a Hypothesis Test Works

• Exact amount Uses sample error

Test hypothesis

How to Test Statistical

Testing a Hypothesis of a Mean

• Example in Text: Rex Reigen

hypothesizes that college interns

make $2,800 in commissions A

survey shows $2,750 Does the

survey sample statistic support or fail

to support Rex’s hypothesis? (p 472)

• Since 1.43 z falls between -1.96z and +1.96 z, we ACCEPT the hypothesis.

How to Test Statistical

s

p

z

H p H

s

x z

H x H

• The probability that our sample mean

of $2,800 came from a distribution of means around a population parameter

of $2,750 is 95% Therefore, we

accept Rex’s hypothesis

Hypothesis Testing

• Non-Directional hypotheses:

hypotheses that do not indicate the

direction (greater than or less than) of

a hypothesized value

Using SPSS to Test Hypotheses

About a Percentage

• SPSS cannot test hypotheses about percentages; you must use the

formula See p 475

Using SPSS to Test Hypotheses

About a Mean

• In the Hobbit’s Choice Case we want

to test that those stating “very likely” to patronize an upscale restaurant are

willing to pay an average of $18 per

entrée

• DATA, SELECT CASES, Likely=5

• ANALYZE, COMAPRE MEANS, ONE SAMPLE T TEST

• ENTER 18 AS TEST VALUE

What if We Used a Directional

Hypothesis?

• Those stating “very likely” to

patronize an upscale restaurant are willing to pay more than an average

of $18 per entrée

• Is the sign (- or +) in the

hypothesized direction? For “more than” hypotheses it should be +; if

not, reject

What if We Used a Directional

Hypothesis?

• Since we are working with a

direction, we are only concerned with one side of the normal distribution Therefore, we need to adjust the

critical values We would accept this

hypothesis if the z value computed is

greater than +1.64 (95%)