This document discusses hypothesis testing, which involves drawing inferences about a population based on a sample from that population. It outlines the key elements of a hypothesis test, including the null and alternative hypotheses, test statistics, critical regions, significance levels, critical values, and p-values. Type I and Type II errors are explained, where a Type I error involves rejecting the null hypothesis when it is true, and a Type II error involves failing to reject the null when it is false. The power of a hypothesis test is defined as the probability of correctly rejecting the null hypothesis when it is false. Controlling type I and II errors involves considering the significance level, sample size, and population parameters in the null and alternative hypotheses.

1. Hypothesis Testing with One Sample

3. James Lind’s experiment

6. Hypothesis testingHypothesis testing
Draw inferences about a population
based on a sample
Testing a claim about a property of
a population

7. Statistical Inference
∗ Inferences about a population are made on
the basis of results obtained from a sample
drawn from that population
∗ Want to talk about the larger population from
which the subjects are drawn, not the
particular subjects!

8. What Do We Test ?
∗ Effect or Difference we are interested in
∗ Difference in Means or Proportions
∗ Odds Ratio (OR)
∗ Relative Risk (RR)
∗ Correlation Coefficient
∗ Clinically important difference
∗ Smallest difference considered biologically
or clinically relevant

9. Example: Gender Selection

12. Hypothesis Testing
Goal: Make statement(s) regarding unknown population
parameter values based on sample data
∗Elements of a hypothesis test:
∗ Null hypothesis - Statement regarding the value(s) of
unknown parameter(s). Typically will imply no association
between explanatory and response variables in our
applications (will always contain an equality)
∗ Alternative hypothesis - Statement contradictory to the null
hypothesis (will always contain an inequality)

13. Null Hypothesis
∗ Usually that there is no effect
∗ Mean = 0
∗ OR = 1
∗ RR = 1
∗ Correlation Coefficient = 0

14. Alternative Hypothesis
∗ Contradicts the null
∗ There is an effect
∗ What you want to prove ?

15. Null Hypothesis expresses no difference
Example:
H0: µ = 0
Often said
“H naught” Or any number
Later…….
H0: µ1 = µ2

16. Alternative Hypothesis
H0: µ = 0; Null Hypothesis
HA: µ = 0; Alternative Hypothesis
Researcher’s predictions should be
a priori, i.e. before looking at the data

17. Estimation: From the Sample
∗ Point estimation
∗ Mean
∗ Median
∗ Change in mean/median
∗ Interval estimation
∗ 95% Confidence interval
∗ Variation

18. Parameters and
Reference Distributions
Continuous outcome data
∗ Normal distribution: N( μ, σ2
)
∗ t distribution: tω (ω = degrees of freedom)
∗Mean = (sample mean)
∗Variance = s2
(sample variance)
Binary outcome data
∗ Binomial distribution: B (n, p)
X

19. Normal Distribution

20. t – Distribution

21. Binomial Distribution

22. Hypothesis Testing
Goal: Make statement(s) regarding unknown population
parameter values based on sample data
Elements of a hypothesis test:
∗ Test statistic - Quantity based on sample data and null
hypothesis used to test between null and alternative
hypotheses.
∗ The test statistic is found by converting the sample statistic
(proportion, mean or standard deviation) to a score (z, tz, t or xx22
))

23. ∗ Critical region (Rejection region): Values of the test
statistic for which we reject the null in favor of the
alternative hypothesis
Critical Region, Significant level,
Critical value and p-value

24. ∗ Significant level (α ): the probability that the test
statistic will fall in the critical region when the null
hypothesis is actually true.
Critical Region, Significant level,
Critical value and p-value

25. ∗ Critical value: is any value that separates the critical
region from the values of the test statistic that do not
lead to rejection of the null hypothesis .
Critical Region, Significant level,
Critical value and p-value

26. ∗ Two tailed: the critical region is in the two extreme
regions (tails) under the curve
Two-Tailed, Left Tailed, Right Tailed

27. ∗ Left tailed: the critical region is in the extreme left
region (tails) under the curve
Two-Tailed, Left Tailed, Right Tailed

28. ∗ Right tailed: the critical region is in the extreme right
region (tails) under the curve
Two-Tailed, Left Tailed, Right Tailed

29. ∗ P-value (p-value or probability value: is the
probability of getting a value of the test statistic that
is at least as extreme as the one representing the
sample data assuming the null hypothesis is true.
∗ The null hypothesis is rejected if the p-value is very
small such as 0.05 or less.
Critical Region, Significant level,
Critical value and p-value

30. Reject the null hypothesis (or other)
Fail to reject the null hypothesis
Prove the null hypothesis to be true
Accept the null hypothesis
Support the null hypothesis
Statistically
correct
Ok but misleading

31. ∗ Traditional Method: Rejection of the null hypothesis
if the statistic falls within the critical region
 Fail to reject the null hypothesis if the test statistic does
not fall within the critical region
∗ P – value methodP – value method: rejection H0 if p-value < α (where α
is the significant level such as 0.05)
Decision Criterion

32. ∗ Another option:Another option: Instead of using a significant level
such as α = 0.05, simply identify the P value and leave
the decision to the reader
∗ Confidence intervals:Confidence intervals: Because a Confidence interval
estimate of the population parameter contains the
likely values of that parameter, reject a claim that the
population parameter has a value that is not included
in the confidence interval
Decision Criterion

33. Statistical Error
Sometimes H0 will be rejected (based on large test
statistic & small p-value) even though H0 is really true
i.e., if you had been able to measure the entire
population, not a sample, you would have found
no difference between and µ some value but
based on X you see a difference.
The mistake of rejecting a true H0 will happen with frequency α
So, if H0 is true, it will be rejected ~5% of the time as α frequently = 0.05

34. 0
0 20
Population mean = 0
Sample mean = 20
Conclude based on sample mean that population mean ≠ 0, but it really
does (H0 true), therefore you have falsely rejected H0
Type I Error
population=“True”
Sample=What you see
H0 : mean = 0

35. Statistical Error
Sometimes H0 will be accepted (based on small test statistic & large
p-value) even though H0 is really false
i.e., if you had been able to measure the entire
population, not a sample, you would have found
a difference between and µ some value- but
based on X you do not see a difference.
The mistake of accepting a false H0 will happen
with frequency β

36. 0
Sample mean = 0
0 20
Sample mean = 20
Conclude based on sample mean that population mean = 0, but it really
does not (H0 really false), therefore you have falsely failed to reject H0
Type II Error
Population= “True”
Sample= what you see
H0 : mean = 0
20

37. 1. The treatments do not differ, and we correctly conclude
that they do not differ.
2. The treatments do not differ, but we conclude that they
do differ.
3. The treatments differ, but we conclude that they do not
differ.
4. The treatments do differ, and we correctly conclude that
they do differ.
Four Possibilities in Testing Whether
the Treatments Differ

39. Type I error
∗ Concluded that
there is difference
while in reality
there is no
difference
∗ α probability
Type II error
∗Concluded that
there is no
difference while in
reality there is a
difference
∗β probability

42. Controlling Type I & Type II Errors
α
β
Power (1 – β)
Sample Size

43. ∗ The power of the hypothesis test is the probability (1-(1- ββ))
rejecting a false null hypothesis, which is computed by
using:
∗ A particular significant level α
∗ Sample size nn
∗ A particular assumed value of the population parameter
in the null hypothesis
∗ A particular assumed value of the population parameter
that is alternative to the value in the null hypothesis
Power of the test

45. Term Definitions
α = Probability of making a type I error
= Probability of concluding the treatments differ when in reality
they do not differ
β = Probability of making a type II error
= Probability of concluding that the treatments do not differ when
in reality they do differ
Power = 1 - Probability of making a type II error
= 1 - β
= Probability of correctly concluding that the treatments differ
= Probability of detecting a difference between the treatments if
the treatments do in fact differ