1. Warm Up
#’s 12, 14, and 16 on p. 552
Then answer the following question;
In a jury trial, what two errors could a jury make?

2. Warm Up – Day 2
In the court system, a suspect is innocent until proven guilty…
Write this in terms of hypotheses and state the Type I and Type II errors

3. 9.1 – Type I & Type II Errors/Sig Tests for Pop Proportion
When we draw a conclusion from a significance test, we hope our conclusion will be correct. But sometimes it will be wrong. There are two types of mistakes we can make.
If we reject H0 when H0 is true, we have committed a Type I error.
If we fail “to” reject H0 when Ha is true, we have committed a Type II error.
Truth about the population
H0 true
H0 false
(Ha true)
Conclusion based on sample
Reject H0
Type I error (α)
Correct conclusion 
Fail to reject H0
Correct conclusion 
Type II error (β)
Try Exercise 39
decrease α  increase β
increase α  decrease β

4. Errors in Trials
You may recall the O.J. Simpson trial. If not, he was found innocent in his
criminal trial but found guilty in his civil trial.
If O.J. is innocent, then an error was made in the civil trial.
If O.J. is guilty, then an error was made in the criminal trial.

5. Type I and Type II Errors
The probability of a Type I error is the probability of rejecting H0 when it is really true…this is exactly the significance level, α, of the test.
Significance and Type I Error
The significance level α of any fixed-level test is the probability of a Type I error.
That is, α is the probability that the test will reject the null hypothesis H0 when H0 is actually true.
Consider the consequences of a Type I error before choosing a significance level.
Try Exercise 39
HINT: A Type II error is when we fail “II” reject H0.
This is not anyone's fault… no mistakes were made in calculations

6. The One-Sample z-Test for a Proportion
To perform a significance test, we state hypotheses (STATE), check conditions (PLAN), calculate a test statistic and P-value (DO), and draw a conclusion in the context of the problem (CONCLUDE).
The four-step process is ideal for organizing our work.
Significance Tests: A Four-Step Process
State: What hypotheses do you want to test, and at what significance level? Define any parameters you use.
Plan: Choose the appropriate inference method. Check conditions.
Do: If the conditions are met, perform calculations.
Compute the test statistic.
Find the P-value.
Conclude: Make a decision about the hypotheses in the context of the problem.

7. The One-Sample z-Test for a Proportion
When the conditions are met—Random, 10%, and Large Counts,
The z statistic has approximately the standard Normal distribution when H0 is true.
P-values therefore come from the standard Normal distribution.

8. Carrying Out a Significance Test
The test statistic says how far the sample result is from the null parameter value, and in what direction, on a standardized scale. (You know this process as finding a z-score) You can use the test statistic to find the P-value of the test.
One Sample z-Test for a Proportion
Choose an SRS of size n from a large population that contains an unknown proportion p of successes. To test the hypothesis H0 : p = p0, compute the z statistic
Find the P-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the alternative hypothesis Ha:
Now complete the Alternate Example: Kissing the right way in your notes packet (p. 99).

9. More Examples
A distributor of eggs has noted that the proportion of eggs that are broken during shipment is 7%. In an attempt to improve, they have designed a new package. In a sample of 500 eggs delivered in the new packaging, only 32 were broken. Does this sample give evidence that the new packaging is better than the old? Explain.
A significance test was performed to test the null hypothesis H0: p = 0.5 versus the alternative Ha: p > The test statistic is z = The P-value for this test is
(a) (c) (d) (e)
Would you reject or fail to reject H0?
(b)
(b)

10. One More…
The water diet requires one to drink two cups of water every half hour from the time one gets up until one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteers agree to test the diet. They are weighed prior to beginning the diet and after six weeks on the diet. For this study, H0: = 0 and Ha: > 0 where is the true mean weight loss.
A Type I error in this setting would mean:
(b) concluding that the diet leads to weight loss when it really does.
(c) not concluding that the diet leads to weight loss when it does.
(d) not concluding that the diet leads to weight loss when it really doesn’t.
(e) drawing a conclusion from this test when the Normality condition has not been satisfied.
(a) concluding that the diet leads to weight loss when it doesn’t.
(a) concluding that the diet leads to weight loss when it doesn’t.