Hypothesis Testing, Intro Statistics,

1. Hypothesis Testing
Foundations I
September 26, 2011

2. Objectives
 Differentiate three measures of central tendency,
including their advantages and disadvantages
 Explain the rationale of hypothesis testing
 Define the null and alternate hypotheses
 Define and interpret: p value, test statistic, type I and II
error, alpha, beta and statistical power
 Explain how statistical power and sample size are
related and describe other factors influencing power
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3. Levels of Measurement
 Categorical (nominal)
 Ordinal
 Interval
 Ratio
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4. Categorical Data
 Non-ordered data
 Often represents different categories: sex, eye
colour, genotypes etc…
 An average would be meaningless
 More meaningful to talk about: different categories,
proportions, percentages or mode
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5. Ordinal Data
 Ordered data
 The distance between the data points may vary
 E.g., Placement in a race, perceived level of pain, or
depression scale
 7 is greater than 5 and greater than 3 but differences
between 7 & 5 may not be the same as 5 & 3
 Average is not meaningful here; finding a middle
number maybe more meaningful and most consistent
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6. Interval Data
 Very similar to ordinal data, but the differences are
consistent
 E.g., Temperature in Celsius or Ferinheight
 Difference between 20 and 30 is the same as the difference
between 40 and 50
 Really well designed rating scales gather interval data
 Important to note that 0 is not meaningful in interval data
 An average (mean) is meaningful unless data is skewed
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7. Ratio Data
 Very similar to interval data except 0 is meaningful
 E.g., Tracking growth of bacteria, height, & weight of babies
 Someone can be twice as tall as another person; however,
cannot say something is twice as hot or cold unless its
measured in Kelvin (in Kelvin temperature of 0 is
meaningful)
 Average is very useful and many statistical procedures for
ratio data are based on means; however, if data is skewed
median is more useful
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8. Central Tendency
 If you wanted to describe a population or a group
of people using one or two numbers you could say:
• On average, students in this class scored about 75% on
last exam….
• In this class, the most frequent eye colour is….
• In a small sub-sample of 10 students, the middle score on
the exam was….
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9. Mean, Median & Mode
 Depending on the type and quality of your data,
either mean, median, or mode may be more
suitable in describing the typical structure of your
data or central tendency
 Statistical analyses such as Analysis of Variance, or
Chi Square Analysis or T-Tests are based on
different measures of central tendency
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10. Descriptive vs. Inferential Statistics
 Descriptive statistics describe the sample or
population usually by providing values of range,
maximum, minimum, central tendency, variance
(sum of individual differences from the mean)
 Inferential statistics are often used when you do
not have access to the entire population and want
to make an inference about this population
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11. A Conjecture…..
 After doing a great deal of reading, the dean of a
well know US medical school believed that in
general, the students in medical programs have an
average IQ of 135
 This is conjecture about an entire population of
undergraduate medical students
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12. Hypothesis Testing: Step 1
 We can test the dean’s conjecture…
Null Hypothesis - Ho: µ=135
Alternative Hypothesis - HA: µ≠135
We test for the conjecture or hypothesis by
making it the null
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13. Role of Software
 Computer programs such as SPSS, SAS, R, STATA,
etc…
 They have built in algorithms to carry out what you
might do by hand
 Its is important to initially do this by hand to
understand what it means to reject, or fail to reject
the null hypothesis
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14. Hypothesis Testing: Step 2
 Because we are not dealing with absolutes and we are
making a prediction about a population its not exact.
 We need to select a criterion or significance level by which
we can either reject or accept the null hypothesis.
 Most often the criterion or significance level is set at .05
 It is also referred to as p-value or α
At what point is the difference between the sample mean
and 135 not due to chance but fact ??
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15. Hypothesis Testing: Step 3
- We sample 10 students
- Area of acceptance is 95%
- Look up critical values on a
t-score table (±2.262)
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16. Hypothesis Testing: Step 4
 We need to randomly draw a sample of 10 Students
115, 140, 133, 125, 120, 126, 136, 124, 132, 129
Mean = 128
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17. Hypothesis Testing: Step 5
 We need to calculate Standard Deviation (SD) &
Standard Error (SE)
 How many people you know has heard of standard
deviation before?
 How many people know what it means?
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18. IQ Scores Mean DiviationsDiviations scores Squared
Scores
115 128 13 169
140
133
128
128
-12
-5
144
25 Before SD we need to
125 128 3 9
120 128 8 64 understand variance
126 128 2 4
136
124
128
128
-8
4
64
16
Standard Deviation – Can
132 128 -4 16 be thought of as an
129 128 -1 1 average of deviation
Sum 0 512
Standard Error – Is an
Sample Variance 0 56.88889
estimation of SD used in
Standard Deviation 0 7.542472 calculating t-statistic
Standard Error 2.385139
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19. T-Test
 The t-statistic was introduced in 1908 by William
Sealy Gosset
 A chemist working for the Guinness brewery in
Dublin, Ireland ("Student" was his pen name)
 Gosset devised the t-test as a way to cheaply
monitor the quality of stout
 Published the test in Biometrika in 1908
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20. Hypothesis Testing: Steps 6 & 7
T-statistic = (sample average – hypothesis)/standard error
t= (128-135)/2.385
t=-2.935
“The hypothesis that the mean IQ
of the population is 135 was
rejected, t= -2.935, df=9, p≤ .05.”
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21. Type I and II Error
 Remember in step 2, we asked how much will we
attribute the difference of means to chance…
 Measurement is never exact; though some journals and
papers vary, a p-value of .05 (meaning that we are 95%
sure that result did not happen by chance) is used
 When we have rejected the null and it is actually true
this is type I error or “false positive”
 When we have not rejected the null and it is actually
false this is a type II error or “false negative”
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22. Power and Measures
 How much power does our prediction have?
 How much can we infer?
 It depends on sample size & quality of the measure
 IQ, Depression Scale, Cognitive ability are unobservable
 Growth of bacteria, cellular effects from medication are
observables – a ruler can be put to it
 The more we can see, the less population we will need
 The more accurate our inferences, the smaller error we
would produce
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23. Contact
 Dr. Saad Chahine
Saad.Chahine@msvu.ca
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