Chapter 16

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 parameters.

The Concepts of Inference and Statistical Inference

• Inference: making a generalization about an entire class (population) based upon what you have observed about a small set of members of that class (sample)

• Statistical inference: a set of procedures in which the sample size and sample statistics are used to make estimates of population parameters

Parameter Estimation

• Parameter Estimation: the process of using sample information to compute an interval that describes the range of values a parameter such as the population mean or population percentage is likely to take on

• Parameter Estimation involves 3 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

Parameter Estimation…continued…Sample Statistic

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…continued…Standard Error

• 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 same sample size, the more variability, the greater the standard error

• 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…continued…Confidence Intervals

• 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…the formula allows the researcher to insert the appropriate Z value representing the desired level of confidence

• 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

Parameter Estimation…cont.

• Five steps involved in computing confidence intervals for a mean or percentage:• Determine the sample statistic.• Determine the variability in the sample for that

statistic.• Identify the sample size.• Decide on the level of confidence.• Perform the computations to determine the

upper and lower boundaries of the confidence interval range.

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 parameter is 41.3% prefer “Rock” music radio stations (n=400) We run FREQUENCIES to learn this

• But, how accurate is this estimate of the true population percentage preferring rock stations?

Parameter Estimation Using SPSS: Estimating a Percentage

• Estimating a Percentage: SPSS will not calculate this 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.

• 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. 468)

Estimating a Population Percentage with SPSS…cont.

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.

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 page 469)

• 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

Estimating a Population Mean with SPSS…cont.

• 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.

Hypothesis Testing

• 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…so we use hypothesis testing in our lives all the time

Hypothesis Testing…cont.

• 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.• 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.

Hypothesis Testing…cont.

• Non-Directional hypotheses: hypotheses that do not indicate a direction (greater than or less than) of a hypothesized value. Rather, non-directional hypotheses state that the hypothesized value is “equal to X.” “Customers expect an entrée to cost $18.”

• Directional hypotheses: hypotheses that indicate the direction in which you believe the population parameter falls relative to some target mean or percentage…”Customers expect an entrée to cost more than $18.”

The Logic of Hypothesis Testing: For Non-directional

hypotheses• IF we ASSUME that the hypothesized value is indeed the

population parameter, then

• 95% of all sample means (or %) drawn from a distribution of sample means (or %) having the value of the parameter will…

• Fall within + or – 1.96 z scores.

• Therefore, if our formula calculates a z score between + or – 1.96, it is likely (95%) that our sample statistic was drawn from a distribution of sample means (or %) around the population parameter we have hypothesized. We accept the hypothesis.

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? (page 476).

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

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.

How Do We Use SPSS to Test Hypotheses About a

Percentage?• SPSS cannot test hypotheses about percentages.

You must use the formula. See page 474 for an example.

How Do We Use 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, COMPARE MEANS, ONE SAMPLE T

TEST• ENTER 18 AS TEST VALUE• Note: z value is reported as t in SPSS output

What if we had stated the hypothesis as 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

• 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%).