9-1 Statistical Tests

1
9-1 Statistical Tests
2
Hypothesis

• Is the global temperature increasing
• Did the laws requiring hands free cell phone use
  result in a decrease in auto accidents?
• Is the housing market truly cooling off?
• These are all hypotheses.. Or suppositions.

3
Hypothesis (cont)

• A Hypothesis proposes a model if the data is
  consistent with the model, then there is no
  reason to think that the hypothesis is false. If
  the facts instead do not fit with the model, then
  perhaps the model might be wrong
• The question is, how far off is acceptable?

4
Hypothesis (cont)

• Hypotheses dont involve estimating population
  parameters but instead are about the
  reasonableness of the value of the parameter.
• Null Hypotheses Ho
• Parameter is correct as stated
• Alternative Hypotheses H1 HA
• Parameter is not correct as stated

5
Hypothesis (cont)

• A Toyota salesman tells you that the Prius gets
  45 mpg. You think he is exaggerating.
• Null Hypotheses Ho µ .75
• Alt. Hypotheses H1(HA ) µ lt .75

6
Hypothesis (cont)

• An allergy drug has been tested and the claim is
  that 75 of patients in a large clinical trial
  find their symptoms significantly reduced.
• Null Hypotheses Ho p po .75
• Alt. Hypotheses H1(HA ) p ? .75

7
Hypothesis (cont)

• Notice that the null is always an equal
  statement, while the alternate will be a greater
  than (right tailed), less than (left tailed) or
  not equal to (two tailed).
• This is important for reading the charts the
  calculator will read the alternate hypothesis
  correctly without you saying left, right etc.

8
Now

• How are we going to decide whether to accept the
  null hypothesis or reject?
• That is, what are we really doing?
• How far off is data from the assumed statistic?
  What is the probability that the observed data is
  realistic with the assumed statistic?

9
Test Statistic

• With normal x and known s, what is the
  probability that the ? value (or z value) exists
  with the assumed µ value?

10
Four Steps

• 1. State the null hypothesis as well as the
  alternate.
• 2. Check the model (normal)
• 3. Calculate the test statistic.
• The goal is to get the P value (the probability
  that the observed statistic value could occur if
  the null hypothesis is correct). The smaller the
  P-value, the more likely the rejection of Ho. It
  suggests that results are less likely due to
  chance.
• 4. State the conclusion. Either reject or fail
  to reject the null hypothesis.

Get in the habit of again drawing pictures to
visualize the test (left, right, two)
11
Mean Example

• Rosie, an aging sheep dog in Montana gets regular
  checkups from the local vet. Let x be a random
  variable that represents Rosies resting heart
  rate (beats per min). From past experience, the
  vet knows that x has a normal distribution with s
  12 and, for dogs of this breed, µ 115.
• Over the past six weeks, Rosies heart rate
  measured an average of 105.0 (six different
  measurements. The vet is concerned that Rosies
  heart rate may be slowing. Do the data indicate
  that this is the case?

12
Steps

• Step 1 Ho µ 115 HA µ lt 115
• Step 2 Independence? Likely. Randomization?
  Nothing indicates anything to the contrary.
• Step 3 find z
• Step 4 As P(? lt 105.0) P(z lt -2.04) 0.0207.
• That is, the probability of getting a sample mean
  below 105.0 is less than 2, so reject Ho and
  accept HA.

-2.04
Note we have NOT proved that the alternate is
true.
13
Types of Errors

• Type 1 Rejecting the null hypothesis when it is
  actually true
• (false positive diagnosing a healthy person
  with a disease, convicting an innocent person)
• Type 2 Accepting the null hypothesis when it is
  false.
• (false negative diagnosing a sick person as
  free from disease, allowing a guilty person to go
  free)

14
Levels of Significance

• a (alpha) probability of rejecting Ho when it
  is true
• i.e. probability of a Type 1 error
• ß (beta) probability of accepting Ho when it is
  false
• i.e. probability of a Type 2 error
• Obviously we want a and ß to be as small as
  possible

15
Levels of significance (cont)
The true situation
Ho true Ho false
Reject Ho Type 1 a Ok
Accept Ho Ok Type 2 ß
What the evaluator does
The power of the test is its ability to detect a
false hypothesis. Power of the test 1 ß The
lower value for ß, the more stringent the
test. The power of the statistical test will
increase as a increases, but a larger value of a
increases the likelihood of a type 1 error.
16
Level of Significance (cont)

• Typically a is decided in advance. Then the
  P-value is determined.
• P-value a then reject the null hypothesis and
  say that the data is statistically significant at
  the given level.
• P-value a then do not reject the null
  hypothesis.

17

• The price to earnings ratio is an important tool
  in financial work. A random sample of 14 large
  US banks gave the following P/E ratios (source
  Forbes)
• 24 16 22 14 12 13 17
• 22 15 19 23 13 11 18
• The sample mean is approximately 17.1. Generally
  speaking, a low P/E ratio indicates a value
  stock. A recent copy of the Wall Street Journal
  indicated that the P/E ratio of the entire SP
  500 stock index is µ 19. Let x b e a random
  variable representing the P/E ratio of all large
  U.S. bank stocks. We assume that x has a normal
  distribution and a s 4.5. Do these data
  indicate that the P/E ratio of all U.S. bank
  stocks is less than 19? Use a 0.05.