AP Statistics

1
AP Statistics

• Hypothesis Test for a Population Mean Cont.

2
Homework Questions?
3
25 Fee Due Thursday!!

• If you are planning on taking the AP test(which
  you all really should do if youre planning on
  going to college) then you need to get this paid.
• Talk to me after class if you want to take the
  test, but finances are an issue.

4
Finding Z Critical Values

• Finding the Z-value that is associated with a
  particular significance level
• Two-Sided Tests
• a .05 ? z 1.96
• a .01 ? z 2.56
• One Sided Tests
• a .05 ? z 1.64
• a .01 ? z 2.33

5
Fixed Significance Level Test

• To test the at a specific confidence level, one
  can simply compare the test statistic z, to the
  Z associated with the significance level of the
  test.
• Generally it is better to compute the p-value and
  that is what we will do.

6
Example Rand Function on TI Calculators

• The rand function randomly generates a decimal
  between 0 and 1.
• The mean of this distribution should be 0.5
• We want to test whether this is true or not.
• First lets set up the Test

7
Rand Function

• Should be sampling from the Uniform distribution
  0, 1).
• s is known, sqrt(1/12) .288675
• We will do 30 trials and determine the mean of
  our sample to see if it differs from 0.5.

8
Hypotheses

• H0 µ .5
• HA µ ? .5
• One sample Z-test of a mean
• Check conditions We have a SRS from the
  calculator, s is known, and n 30 trials is
  large.
• Lets run the trial with a .05

9
Seed the rand function, then run trial

• (last 4 digits of phone ) sto-gt rand
• (Rand is under math, prob, 1.
• Seq(rand, X, 1, 30, 1) sto-gt L1
• Seq is under 2nd list, OPS, 5
• X, 1, 30, 1 means run for x 1 to 30,
  incrementing at 1 each time.
• Then 1 var stats on L1 to get our data.

10
Test Statistic, Curve, P-value

• Do this for your own data.

11
Conclusion in Context.

• Now, a fixed level two sided test for a .05
  would have Z at 1.96, so if you z test statistic
  was beyond 1.96, you would reject H0 in favor of
  HA.

12
Confidence Intervals and 2-sided tests

• There is a close connection between CIs and
  2-sided tests. In fact, if the 95 CI does not
  include µ0, then the a .05 hypothesis test will
  reject µ0.
• (This is true for any a) 99 ? a .01, etc.

13
Practice
14
Exercises

• p581 10.43, 10.44, 10.45

15
Thinking Deeper about Significance Tests

• The context of a problem can dictate the
  significance level.
• a .05 is not universal.

16
Practical Significance versus Statistical
Significance

• Example The healing of small cuts has mean
  healing time 7.6 days with standard deviation 1.4
  days.
• A new cream is used in a study of healing times
  and generates a sample mean of 7.1 days.
• Lets quick do the test on the calculator.
• Just because the cream is better, doesnt mean we
  care. Whats 12 hours anyway?

17
Bad Data

• Poorly designed experiments can generate all
  kinds of results, but that doesnt mean that
  theyre any good.

18
Multiple Analyses

• Example The military runs 100 statistical tests
  to determine which height, weight, cholesterol
  levels, body fat percentages, hair color, income
  level, education level, SAT score, etc.
  correspond to making good soldiers.
• They determine that short, blond soldiers are the
  best.
• If you run 100 tests at a .05, you would expect
  at least 5 to reject just on chance, so you
  should be wary of doing tests at random. An
  intentionally and carefully constructed study is
  the standard.

19
Exercises

• p589 10.58
• p592 10.61, 10.65
• Read through section 10.3 Making sense of
  statistical Significance.

20
Type I Error

• Statistical Inference is not perfect. Even with
  a .01, we still expect to make a mistake 1 out
  of 100 times.
• A Type I error is associated with a.
• It occurs when you reject H0 even though H0 is
  true.
• In practice, youll never know whether or not you
  made a type I error or not.

21
Type II Error

• A type II error is a little trickier. Its what
  occurs when you should have rejected H0, but you
  didnt have conclusive evidence so you didnt.
• Perhaps you got a p-value of 0.07, so you failed
  to reject H0, but in fact HA is true.

22
Type I and Type II Errors
23
Calculating Probabilities

• a is the probability of making a type I error.
• ß is the probability of making a type II error.
• ß is a little more difficult to calculate.
• (In fact its impossible without knowing the true
  µ, but we can calculate it for various what if
  scenarios.

24
Ways to Decrease a and ß