Download Statistics 400 - Lecture 2

Statistics 400 - Lecture 10
 Last day: 8.3 and started 8.4
 Today: Sections 8.4
Hypothesis Testing
 Hypothesis testing is a statistical technique to test if a conjecture
about a population parameter is true
 Has 4 Main Steps:
 Null and Alternate Hypotheses
 Test Statistic
 P-Value
 Decision based on pre-specified error rate
Example
 Heights of one-year-old girls normally distributed with mean 30
inches and standard deviation of 1.2 inches
 Company claims taking 500 mg of Vitamin C makes the girls taller
 Is the company’s claim true?
1. Hypotheses
 Hypotheses are statements about a population and is expressed in
terms of the population parameters
 Begin by making an assumption of no change
 (Treatment has no effect)
 This statement is called the null hypothesis (H0)
 Test will be designed to assess evidence against H0
 Hypothesis we suspect is true is called the alternate hypothesis
(H1 )
 Assume H0 is true, collect data and see if there is evidence against
H0 and in favor of H1
Example
 Heights of one-year-old girls normally distributed with mean 30
inches and standard deviation of 1.2 inches
 Company claims taking 500 mg of Vitamin C makes the girls taller
 H0:
 H1:
2. Test Statistic
 Test statistic measures compatibility between H0 and the data
 It is based on 2 principles:
based on estimate of the parameter that appears in the
hypotheses
measures distance of estimate from the hypothesized value
 When H0 is true, we expect the value of estimate to be close to
parameter on average
Example (continued)
 Suppose a random sample of 100 baby girls are given 500 mg of
vitamin C daily for 1 year
 Mean height of the girls after 1 year is 32 inches (estimates
population mean)
 What is the distribution of
x if H
0
is true?
 What is the distribution of
x if H
1
is true?
3. P-Value
 Assume that H0 is true
 The P-value is the the probability of observing a test statistic
as extreme or more extreme than the value actually
observed when H0 is true
 What does a small p-value imply?
 How small is small?
Example (continued)
 If H0 is true, the distribution of the sample mean is:
 What does “extreme” mean in this context?
 P-value=
4. Decision
 How small must the p-value be to reject H0?
 Must decide which value of the test statistic give evidence in favor
of H1
 Would like the probability of observing such values to be small
when H0 is true
 The significance level of the test is:
Example (continued):
 P-value=
 Significance level:
 Decision:
Hypothesis Testing is Similar to a Jury Trial
 H0: state of no change
 Not Guilty
 H1: condition believed to be true
 Guilty
 Collect data and compute test
statistic
 Collect evidence
 Compute p-value
 Weigh evidence
 Reject or do not reject H0 based
on significance level
 Decide if evidence is in favor of
guilty beyond a reasonable doubt
 How do we interpret significance level
 Some common significance levels:
 Have we proven that H0 is true or false?
Z-Test for the Population Mean
 Have a random sample of size n ; x1, x2, …, xn

H 0 :   0
 Test Statistic: Z 
X 
S/ n
 Can be used for normal population or for large samples (why?)
Z-Test for the Population Mean (cont.)
 P-value depends on the alternative hypothesis:
H1 : 
 0 : p - value  P(Z  z)
H1 : 
 0 : p - value  P(Z  z)
H1 : 
 0 : p - value  2P(Z  | z |)
Example:
 Scientists believe that abused children show elevated levels of
depression
 To test this assertion, as random sample of 50 abused children were
given a Profile of Moods States (POMS) test
 The results showed a mean depression score of 17.3 and standard
deviation of 5.4
 Test, at the 5% level, whether abused children have a higher mean
depression that that of the general population (mean=15)
Example:

A study titled “St. John’s Wort: Effect on CYP3A4 Activity” (Clinical Pharmacology and
Therapeutics, 2000) reported a study that assesed urinary 6-beta-horoxycortisol/cortisol ratio in
12 subjects after 14 days of therapy with St. John’s Wort.

The baseline mean ratio for the target population is 7.0 and the scientists wished to determine if
the therapy resulted in increased a urinary 6-beta-horoxycortisol/cortisol ratio

Using the data below, test this hypothesis
Patient
urinary 6-betahoroxycortisol/cortisol
ratio
Patient
urinary 6-betahoroxycortisol/cortisol
ratio
1
2
3
4
5
6
16.8
13.7
11.3
20.3
7.0
6.1
7
8
9
10
11
12
5.4
14.9
9.2
6.4
12.9
7.2