|
|
|
Hypothesis Testing
Lecture Goals You should be able to: - Understand the relationship between sampling error and hypothesis testing - Test hypotheses - Explain the logic
of testing hypotheses
TBA
Definitions hypothesis = a prediction about a change Everyday hypotheses: Flowers and candy Seasonal temperatures
= assumption of status quo = nothing unusual has occurred = assumption that something has changed null hypothesis = assumption of status quo = assumption of innocence = nothing unusual has occurred
alternative hypothesis = assumption of change (not innocent) Collect data (evidence) Observe evidence (data)
Do data provide reason to reject the assumption of innocence? - No, assume null hypothesis is true (not guilty verdict ) - Yes, assume opposite of null hypothesis is true (alternative hypothesis = guilty verdict ) Start by assuming null hypothesis is correct (assumption of fair coin) Collect data (evidence) Observe evidence (data)
Q = Are the results likely if the coin is fair? A = Yes - Assume null hypothesis is true (fair coin)
Q = Are the results likely if the coin is fair? A = No Why did we get these results? 1) an unlikely event occurred 2) our initial assumption is wrong Odds of # 1 = .0001, # 2 more probable
Results: 10 heads & 0 tails Q = Are the results likely if the coin is fair? A = No
2) our initial assumption is wrong - assumption of fair coin wrong - null hypothesis wrong - reject null hypothesis - assume opposite is true - assume alt. hyp. is correct - conclude coin is NOT a fair coin
Psychological Research hypothesis = a prediction about a change Usually about group differences - 2 different groups (treatments) - control vs. experimental groups
alternative hypothesis = assumption of change/effect = assumption of group differences
Positive reinforcement promotes learning Women mature faster than men
null hypothesis = assumption of no change/effect = assumption of no group differences Women and men mature at equal rates
Formal Process Outline Collect data (evidence) Observe evidence (data) State alternative hypothesis Set level of significance Calculate the z-score Find percentile Decision about the null hypothesis Draw a conclusion The IQ scores of all Americans are normally distributed with a m = 100 and s = 16. The researcher randomly selects one
PSU student and finds that student's IQ score is 130.
Step 1: State the Null Hypothesis null hypothesis = assumption of no change/effect = assumption of no group differences = assumes all scores from SAME Ho : mean IQ of PSU students = mean IQ Ho : m
psu = 100 (or m
psu = m
Amer)
Step 2: State the Alternative Hyp. alternative hypothesis = assumption of group differences = assumes scores from DIFFERENT
HA : mean IQ of PSU students ¹ mean IQ HA : m
psu ¹
100 (or m psu
¹ m
Amer)
Time Out for Theory Q = Are the numbers from the same population of scores or different populations of scores? Q = How likely is it that the scores are from the same population of scores? A = Depends on sampling error & criterion that an unlikely event has occurred HO = assumption of no group differences Assumes that distribution of PSU IQ scores and the distribution of American IQ scores have identical characteristics Means that randomly selected PSU score is from a normal distribution with m = 100 and s = 16 What is the likelihood that the score is 1s
from the mean or closer?
(graph)
What is the likelihood that the score is between
1s
and 2s from
the mean?
(graph)
What is the likelihood that the score is between
2s
and 3s from
the mean?
(graph)
What is the likelihood that the score is greater
than 3s
from the mean?
(graph)
The researcher randomly selects one PSU student and finds that student's IQ score. Distance from the mean Likelihood less than 1s 68.14% 1s to 2s 27.08% 2s to 3s 4.30% greater than 3s
.26%
More scores are close to the mean Fewer scores are farther from the mean More likely to randomly select scores closer to the
mean
Less likely to randomly select scores farthest from the mean Randomly selecting a score that is far from the mean
is an unlikely event
Sampling Error When the score is close to the mean, the sampling error is small More likely to randomly select scores closer to the mean - More likely to have small sampling error
Less likely to randomly select scores farthest from the mean - Less likely to have large sampling error
Large sampling error is an unlikely
event
Sampling Error If we get an unlikely result? 1) unlikely event occurred by chance **2) our initial assumption is wrong
What do we do? - we assumed HO (PSU score from A. pop.) - reject null hypothesis - assume opposite is true (HA) - conclude mean IQ score of PSU students is different
from the mean IQ score of all Americans
Step 3: Set Level of Significance Larger sampling errors = more unlikely Larger z scores = more unlikely How large does a z score have to be to be considered
unlikely?
Set arbitrarily a = symbol for level of significance = z-scores in the outer 5% or outer 1% of the distribution are considered unlikely so we should reject HO a = .01 Means we decide
z-scores in the outer 1% of distribution are unlikely
3 Types of Hypotheses: 1) Group 1 scored higher than Group 2 2) Group 1 scored lower than Group 2 3) Group 1 and Group 2 scored differently a = .05 = Z-scores in
extreme 5th percentile are unlikely
1) Group 1 scored higher than Group 2 a = .01 = Z-scores in
extreme 1 percentile are unlikely
a = .05 = Z-scores in
extreme 5th percentile are unlikely
2) Group 1 scored lower than Group 2 a = .01 = Z-scores in
extreme 1 percentile are unlikely
a = .05 = Z-scores in
extreme 2.5th percentiles are unlikely
Step 3: Set
Level of Significance
3) Non-directional difference a = .01 = Z-scores in
extreme .005th percentiles are unlikely
Step 4: Calculate Z-Score
(see table) Hypothesis is non-directional (formula)
(formula)
Use Area C to find percentile rank for score Area for Z1.875 = .0310 Indicates that score is in the extreme 3rd percentile
Step 6: Decision About Ho For a = .05 = Z-scores in extreme 2.5th percentiles are unlikely Calculated score is in extreme 3rd percentile, does
NOT qualify as unlikely
What if research questions was different? A researcher wants to know whether the mean IQ score
of PSU students is GREATER than the mean IQ
score of all Americans
Calculated score is in extreme 3rd percentile, DOES
qualify as unlikely
When calculated percentile £ a reject Ho When calculated percentile > a fail to reject Ho
Symbols: p = symbol for calculated percentile If p > a
fail to reject Ho
If p > a
fail to reject Ho
If p > a
fail to reject Ho
¹ necessarily mean person is innocent = not enough evidence to convict = not enough evidence to overcome presumption of innocence = not enough evidence to reject Ho (our original assumption) m PSU > m AMER m PSU <
m AMER
If p > a fail to reject Ho If p > a /2 fail to reject Ho p = .0310 If p > a
/2 fail to reject Ho
|
