Download Tests of Significance

The manipulation of statistical formulas is no substitute for knowing what one is
doing. Hubert M Blalock Jr
Chapter 10 - Sec 10.2
The second type of statistical inference about a population based on information from a
sample is "TEST of SIGNIFICANCE". These tests have a goal of assessing evidence
provided by the data about "some claim" about the population.
Claim: "I make 80% of my free throw attempts in basketball."
"Cola drinks lose sweetness over time."
"Students over the age of 30 have better attitudes toward school."
Validating these statements (or refuting them) is done through a significance test.
Significance tests answer two questions:
a) Do the sample results (however small) reflect the true parameter?
b) Would the outcome result easily be explained by chance?
Procedures:
1) Identify the population of interest and the parameter you want to draw conclusions
about. (μ = ___ or p = ___ )
2) State the null and alternative hypotheses in words and symbols. (H0 and Ha)
3) Verify conditions/state assumptions (SRS, pop. at least 10n, approx. normal)
4) Name the procedure (One sample mean or prop. z test)/write the formula, find pvalue, reject or fail to reject Ho
5) Interpret your results in the context of the problem. (There is/is not sufficient evidence
to suggest… )
The NULL HYPOTHESIS
(H0 pronounced "H-nought") declares that there is NO effect or change in the population.
The ALTERNATE HYPOTHESIS
(Ha) declares that there is some effect or change in the population.
A significance test works by asking..."How unlikely is the observed outcome if the null
hypothesis were really true" by assigning a number called the P-value which designates
the probability that the Null Hypothesis is true).
Remember "p" for probability. The p values are calculated from Table A (z-scores)
Generally, we begin by assuming that the null hypothesis is TRUE and attempt to
disprove it by finding a very low likelihood that it is true.
LOGIC: IF the probability of a result is very low then the result is SURPRISING and
provides strong evidence against the original assumption (hypothesis).
**Small p values are evidence against the Null Hypothesis because they say that
observed result is unlikely to occur just by chance. Large p-values fail to give evidence
against the Null Hypothesis.
How small must a p-value be to persuade us to reject the null hypothesis???
Rule of Thumb: p-value less than .05 is called STATISTICALLY SIGNIFICANT.
Let's see how all this works.
Example 10.9
Diet colas use artificial sweeteners to avoid sugar. Colas with artificial sweeteners
gradually lose sweetness over time. Manufacturers therefore test new colas for loss of
sweetness before marketing them. Trained tasters sip the cola along with drinks of
standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then
stored for a period of time, then each taster scores the stored cola. This is a matched pairs
experiment. The reported data is the difference in tasters’ scores. The bigger the
difference, the bigger the loss in sweetness.
2.0
2.2
0.4
-1.3
0.7
1.2
2.0
1.1
-0.4
2.3
The sample mean x  1.02 indicates a small loss of sweetness.
Consider that a different sample of tasters would have resulted in different scores, and
that some variation in scores is expected due to chance.
Does the data provide good evidence that the cola lost sweetness in storage?
To answer that question, we will perform a significance test.
1.
Identify the parameter.
The parameter of interest is μ, the mean loss in sweetness.
2.
State the null hypothesis.
There is no effect or change in the population. This is the statement we are trying
to find evidence against.
The cola does not lose sweetness. H0: μ = 0
State the alternative hypothesis.
There is an effect or change in the population.
This is the statement we are trying to find evidence for.
The cola does lose sweetness. Ha: μ > 0
3.
Verify conditions.
Assume an SRS, approx. normal, pop at least 10n
Suppose the individual tasters’ scores vary according to a normal distribution with mean
μ and σ = 1. We want to test the null hypothesis so we assume μ = 0.
So the sampling model for x is approximately normal with mean μ = 0 and standard

1
deviation

 0.316
n
10
N(0, 0.316)
4.
-0.9 -0.6 -0.3 0
0.3 0.6 0.9
Calculate a statistic to estimate the parameter.
Is the value of the statistic highly unlikely? If so, reject the null hypothesis. If
not, fail to reject the null hypothesis.
Calculate the P-value.
If it is small, your result is statistically significant.
Our sample mean, x , was 1.02. Assuming that the null hypothesis is true, what is the
probability of getting a result at least that large?
1.02  0
z
 3.23
p-value = 1 - 0.994 = .0006
.0006 < .05
.316
STAT – TESTS – 1:Z Test - Calculate
Reject H0
The P-value is 0.0006, meaning that we would only expect to get this result in 6 out of
10,000 samples. This is very unlikely, so we will reject the null hypothesis in favor of the
alternative hypothesis and conclude that the cola actually did lose sweetness.
If the P-value is small we say that our result is statistically significant. The smaller the Pvalue, the stronger the evidence provided by the data.
How small is small enough? Compare the P-value to the value of the significance level
.05. This value is usually predetermined. If the P-value is as small as or smaller than .05,
we say that the data are statistically significant at level .05.
5.
Interpret your results.
There is sufficient evidence to suggest the true loss of sweetness is greater than 0.
Hypotheses can be one-sided or two-sided.
▪ Ha: μ > 0 is a one-sided hypothesis because we are only looking at one direction,
greater than.
▪ Ha: μ ≠ 0 is a two-sided hypothesis because we are looking at two directions,
greater than and less than.