1.
Chapter 16
Generalizing a Sample’s
Findings to Its
Population and Testing
Hypotheses About
Percents and Means
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.
3.
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
4.
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
5.
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
6.
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.
9.
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
10.
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.
11.
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?
12.
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)
13.
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.
14.
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
15.
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.
16.
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
17.
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.
18.
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.”
19.
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.
20.
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).
21.
Since 1.43z falls between -1.96z and +1.96 z, we
ACCEPT the hypothesis
22.
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.
23.
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.
24.
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
25.
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%).
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