Outline

1
Outline

• 1. Hypothesis Tests Introduction
• 2. Technical vocabulary
• Null Hypothesis
• Alternative Hypothesis
• a (alpha)
• ß (beta)
• 3. Hypothesis Tests Format
• 4. Hypothesis Tests - Examples

2
Hypothesis Tests - Introduction

• Hypothesis tests ask questions about population
  means (or variances).
• These are always questions about probability.
• E.g., if population mean on some dimension is ?0,
  what is the probability that a random sample
  drawn from that population would have a sample
  mean in some range relative to ?0?

3
Hypothesis Tests - Introduction

• If that probability turns out to be very small,
  we conclude that the sample was not drawn from
  the population with mean ?0.
• This is useful when ?0 is the mean for an
  untreated population and is the mean for a
  treated sample.
• In that case, we conclude that the treatment had
  an effect (i.e., that ?T ? ?0)

4
We wouldnt be surprised to find in this range
here.
X
?o
L
We would be surprised to find X in this range
here, if µO is true.
5
Hypothesis Tests - Introduction

• The logic
• either something unlikely happened, or
• our sample was not drawn from the population
  with mean ?0 that is, the treatment worked
• so, which is it?

6
Hypothesis Tests - Introduction

• Did something unlikely happen? Is that why our
  sample mean is so far from µO?
• Probably not.
• If unlikely is sufficiently unlikely, we
  conclude that the treatment worked. µT is really
  different from µO.

7
Hypothesis Tests technical vocabulary

• Null Hypothesis (H0)
• Hypothesis of no effect that the treatment
  didnt work
• Hypothesis that the historical population mean
  ?0 is still true.
• Alternative Hypothesis (HA)
• Hypothesis that the treatment worked so
  treated population mean ?A ? ?0.

8
Hypothesis Tests technical vocabulary

• a (Alpha)
• Probability of rejecting the null hypothesis
  when you should not
• Recall you have to choose did treatment work,
  or did something unlikely happen? a measures how
  unlikely.
• with a .05, if you repeatedly sampled from the
  population with mean ?0, you would get a sample
  mean that far from ?0 lt 5 of the time

9
Hypothesis Tests technical vocabulary

• Using a .05, has to be so far from ?0 that
  it would occur by chance 5 times if you
  selected and measured 100 random samples.
• But of course 5 can be gt 0.
• So, sometimes you will conclude that there was a
  treatment effect when in fact you just got one of
  those outlier samples.
• Then, you make a mistake in rejecting H0. That
  mistake is called a Type I error.

10
Hypothesis Tests technical vocabulary

• Question why settle for 5 chances in 100 of
  making a Type I error? Why not go for 1 chance in
  100? Or 0 chances in 100?
• Answer because as we move the critical value of
  
• out into the tail of the distribution around
  ?0, we increase the probability of Type II error
  not rejecting H0 when we should.

11
a
L
?0
might be below the critical value L even if
the treatment worked
?A
ß
12
µO
µA
13
Hypothesis Tests technical vocabulary

• ß (Beta) is the probability of a Type II error
  not rejecting H0 when you should.
• In order to compute ß, you must have a specific
  alternative hypothesis.
• On previous slide, this was shown as ?A.
• In order to compute ß you have to do two steps
  (1) compute L (2) compute probability that
  is smaller (or larger) than L given ?A.

14
a
L
?0
1. Given a, compute L. 2. Knowing L, compute ß.
?A
ß
15
Hypothesis Tests - Format

• H0 ? ?0 (this is the historical population
  mean)
• HA ? lt ?0 HA ? ? ?0
• or HA ? gt ?0
• (One-tailed test) (Two-tailed test)
• Test Statistic Z - ? ? - ?
• ? s

16
Hypothesis Tests - Format

• Rejection Region
• One-tailed test Two-tailed test
• Zobt gt Za Zobt gt Za/2
• or Zobt lt -Za
• Always report your decision explicitly! (Did you
  reject H0 or not reject H0?)
• You NEVER accept H0 only fail to reject it.

17
Hypothesis Tests Example 1

• The average length of classical music CDs is
  known to be 70.6 minutes. You suspect pop music
  CDs are shorter. To test this hypothesis, you
  test a random sample of 100 pop music CDs. Shown
  below are some calculations on the data you
  collect.
• Sx 6250 Sx2 479725
• a. What should you conclude about the relative
  lengths of classical and pop music CDs? (a .01)

18
Hypothesis Tests Example 1

• H0 ? 70.6
• HA ? lt 70.6
• Test Statistic Z - ?0
• SX
• Rejection region Zobt -2.33

Why one-tailed?
Why SX? Why not S?
Why negative?
19
Hypothesis Tests Example 1

• Zobt 62.5 70.6
• 30/v100
• -2.7
• Decision Reject H0. There is evidence that pop
  music CDs are shorter than classical music CDs.

Why 30?
20
Hypothesis Tests Example 2

• At this time of year, half of the tomatoes at
  Fonzies Fine Foods have a shelf life of 5 days
  or more once they arrive at the store. Fonzie is
  considering a new type of genetically-modified
  tomato, hoping these will have a longer shelf
  life. He plans to order a sample of 25 of these
  new tomatoes and will conclude that they last
  longer if 19 or more of them stay fresh for at
  least 5 days on the shelf.

21
Hypothesis Tests Example 2

• a. Suppose that these genetically-modified
  tomatoes, in fact, do not last any longer than
  the other tomatoes Fonzie was buying. What is the
  probability that he will incorrectly conclude
  that the new tomatoes do have a longer shelf life?

22
Hypothesis Tests Example 2

• a. P(tomato stays fresh 5 days) .5
• We want P(X 19 p .50) when n 25.
• That value 1 P(X 18p .50) when n 25.
• From table, p .993. So the probability of an
  incorrect conclusion is 1 - .993 .007

23
Hypothesis Tests Example 2

• b. Suppose that Fonzie buys a case of 50 of the
  new genetically-modified tomatoes and tests the
  hypothesis that these tomatoes will remain fresh
  longer than 5 days using a .05. What is the
  probability that he will conclude that the new
  tomatoes do last longer if the new tomatoes
  actually last for 8 days, on average, with a
  standard deviation of 8.75 days?

24
Hypothesis Tests Example 2

• b. This is a question about Power.
• The power of a test is the complement of ß.
• ß is the probability that you do not reject H0
  when you should.
• Power is the probability that you do reject H0
  when you should (power 1 ß)

25
a
L
5
C
1. Given a, compute L. 2. Knowing L, compute
C. 3. Add .5 C to get power.
8
ß
26
Hypothesis Tests Example 2

• a .05. Therefore Z 1.645.
• L ?0 Za(SX/vn)
• L 5 1.645 (8.75/v50)
• 5 2.036
• 7.036

27
a
5
C
7.036
8
ß
28
Hypothesis Tests Example 2

• Now, find area C
• Z 7.036 8
• (8.75/v50)
• Z -0.78
• Associated p .2823 (from Table).

29
a
5
.2823
1. Given a, compute L. 2. Knowing L, compute
C. 3. Add .5 C to get power.
.5
8
ß
30
Hypothesis Tests Example 2

• The power of this test is .2823 .5 .7823.
• Note that this is the power of the test
  calculated for the specific HA that ? 8.
• If HA specified a different value of ?, this
  procedure would produce a different value for the
  power of the test.

31
Hypothesis Tests