Generalizing a Sample’s Findings to

Its Population and Testing Hypotheses About Percents and

Means

Ch 16 2

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 parameters.

Ch 16 3

Ch 16 4

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

Ch 16 5

Ch 16 6

How to Calculate Sample Error (Accuracy)

n

pqzerror

sp

Where z = 1.96 (95%) or 2.58 (99%)

Sample Size and Accuracy

0%2%4%6%8%

10%12%14%16%

Sample Size

Acc

urac

y

Ch 16 7

Accuracy Levels for Different Sample Sizes

• At 95% ( z = 1.96)• n p=50% p=70% p=90%

• 10 ±31.0% ±28.4% ±18.6%• 100 ±9.8% ±9.0% ±5.9%• 250 ±6.2% ±5.7% ±3.7%• 500 ±4.4% ±4.0% ±2.6%• 1,000 ±3.1% ±2.8% ±1.9%

The “p” you found in your sample

1.96 times sp

95% Confidence interval: p ± 1.96 times sp

Ch 16 8

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

Ch 16 9

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 if we were to repeat the study many times over

Ch 16 10

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

Ch 16 11

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.

Ch 16 12

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.

Ch 16 13

Standard Error of the Mean

Ch 16 14

Standard Error of the Percentage

Ch 16 15

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

Ch 16 16

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.

Ch 16 17

• Theoretical notion

• Take many, many, many samples

• Plot the p’s

• 95 % will fall in confidence interval

(p ± z times sp)

How do I interpret the confidence interval?

2.5%2.5%

95%

Ch 16 18

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

Ch 16 19

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

Ch 16 20

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, n=400, 95%=1.96, calculate Sp.

• 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 46.1%. (See p. 464).

Ch 16 21

How to Compute a Confidence Interval for a Percent

n

pqzsp

• Determine the confidence interval using

• Sample size (n)

• 95% level of confidence (z=1.96)

• P=?%; q=100%-?%

n

pqzp n

pqzp

Lower boundary Upper boundary

Ch 16 22

Estimating a Population Percentage with SPSS

• How do we interpret the results?– Our best estimate of the

population percentage that prefers “Rock” radio is 41.3 percent, and we are 95 percent confident that the true population value is between 36.5 and 46.1 percent.

Ch 16 23

Parameter Estimation Using SPSS: Estimating a Mean

• SPSS will calculate a confidence interval around a mean sample statistic.

• From the Hobbit’s Choice data assume– We want to know how much those

who stated “very likely” to patronize an upscale restaurant spend in restaurants per month. (See p. 465.)

Ch 16 24

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.

Ch 16 25

Ch 16 26

Ch 16 27

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 the formula p±1.96 Sp.

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

Ch 16 28

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?

Ch 16 29

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 forecasting model “passes” the test.

Ch 16 30

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

Ch 16 31

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

Ch 16 32

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

Ch 16 33

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 (known as supported).

• If your hypothesis is incorrect, it will fall outside the confidence interval (known as not supported)

Ch 16 34

How a Hypothesis Test Works

• Sample ----- Population

• Exact amount---- Uses sample error

• percent ----- Test against Ho

• average ----- Test against Ho

Test hypothesis

Ch 16 35

How to Test Statistical Hypothesis

2.5%2.5%

95%

-1.96+1.96

Ch 16 36

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)

Ch 16 37

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

Ch 16 38

How to Test Statistical Hypothesis

2.5%2.5%

95%-1.96 +1.96

npq

p

s

pz

H

p

H

Supported

Not Supported Not Supported

n

sx

s

xz

H

x

H

Ch 16 39

• 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.

Ch 16 40

Hypothesis Testing

• Non-Directional hypotheses: hypotheses that do not indicate the direction (greater than or less than) of a hypothesized value

Ch 16 41

Hypothesis Testing

• Directional hypotheses: hypotheses that indicate the direction in which you believe the population parameter falls relative to some target mean or percentage

Ch 16 42

Using SPSS to Test Hypotheses About a Percentage

• SPSS cannot test hypotheses about percentages; you must use the formula. See p. 475

Ch 16 43

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• Note: z value is reported as t in output.

Ch 16 44

Ch 16 45

Ch 16 46

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.

Ch 16 47

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%).

Ch 16 48