We looked at screen tension and learned that when we measured the screen tension of 20 screens that the mean of the sample was 306.3. We know the standard deviation is 43.

1

• We looked at screen tension and learned that when
  we measured the screen tension of 20 screens that
  the mean of the sample was 306.3. We know the
  standard deviation is 43.
• Find an 80 confidence interval for µ.
• Find a 99.9 confidence interval for µ.
• How large a sample would you need to produce a
  95 confidence interval with a margin of error no
  more than 3?

2
Section 9.1Introduction to Significance Tests
3
A Rose By Any Other Name

• Significance Tests go by a couple of other names
• Tests of significance
• Hypothesis Tests

4
Inference

• So far, weve learned one inferential method
  confidence intervals. Confidence intervals are
  appropriate when were trying to estimate the
  value of a parameter.
• Today, well investigate hypothesis tests, a
  second type of statistical inference. Hypothesis
  tests measure how much evidence we have for or
  against a claim.

5

• A significance test is a formal procedure for
  comparing observed data with a claim (also called
  a hypothesis) whose truth we want to assess. The
  claim is a statement about a parameter, like the
  population proportion p or the population mean µ.
  We express the results of a significance test in
  terms of a probability that measures how well the
  data and the claim fit.agree.

6
The Reasoning Behind Tests of Significance

• I say I am an 80 free throw shooter. You say
  PROVE IT!
• So, I shoot 25 free throws and only make 16. You
  say that Im a liar.
• Your reasoning is based on how often I would only
  make 16 or fewer free throws if I am indeed an
  80 free throw shooter. In fact, this probability
  is 0.0468. The small probability of this
  happening convinces you that my claim was false.

7
(No Transcript)
8
Diet Colas

• Diet colas use artificial sweeteners, which lose
  their sweetness over time. Manufacturers test
  new colas for loss of sweetness before marketing
  them. Trained testers sip the drink and rate the
  sweetness on a scale from 1 to 10 (with 10 being
  the sweetest). The cola is then stored, and the
  testers test the colas for sweetness after the
  storage. The data are the differences (before
  storage after storage), so bigger numbers
  represent a greater loss of sweetness.
• This is a matched pairs experiment!

9
The Data
2.0 0.4 2.0 -0.4 2.2
-1.3 1.2 1.1 0.7 2.3
Most of the numbers are positive, so most testers
found a loss of sweetness. But the losses are
small, and two of the testers found a GAIN in
sweetness. So do these data give good evidence
that the cola lost sweetness in storage? Start
by finding x-bar.
10
Heres our question

• The sample mean is 1.02. Thats not a large
  loss. Ten different testers would likely give
  different sample results.
• Does the sample mean of 1.02 reflect a REAL loss
  of sweetness? OR
• Could we easily get the outcome of 1.02 just by
  chance?

11
Hypotheses

• We will structure our test around two hypotheses
  about the PARAMETER in question (in this case,
  the parameter is µ, the true mean loss of
  sweetness for this cola.)
• The two hypotheses are
• the null hypothesis (no effect or no change)
  represented by H0 (H-naught)
• the alternative hypothesis (the effect we
  suspect is true) represented by Ha.

12
For the Cola problem

• In words, what is the null hypothesis?
• In words, what is the alternative hypothesis?

13
Our Hypotheses in symbols

• In this example,

14
What is a p-value?
Weve found p-values before. P-values are the
area of the shaded region in our normal curve.
Now that area has a name!!!

• A p-value is the probability of getting a sample
  result as extreme or more extreme given that the
  null hypothesis is true.
• The smaller the p-value is, the more evidence we
  have in favor of Ha, and against Ho.
• If the p-value is low (standard is less than
  .05), reject the Ho!
• When the p-value is low, we say the results are
  statistically significant.

15
Back to the cola

• Find the P-value for the problem. Note We know
  that the standard deviation is 1.
• We find that the P-value is 0.0006.
• This means that only 6/10,000 trials would result
  in a mean sweetness loss of 1.02 IF the true mean
  is zero. Since this is so unlikely to happen,
  you have good evidence that the true mean is
  greater than zero.

16

• P - parameters
• H - hypotheses
• A - assumptions
• N - name your test
• T - find your test statistic
• O - obtain your p-value
• M - make a decision (reject or fail to reject)
• S - state a conclusion in the context of the
  problem

17
Types of Alternate Hypotheses and Their Graphs
These are called one-sided alternatives. You
only shade one side. It is either greater than
or less than.
This is called a two-sided alternative. You
shade two sides. It could be less than or
greater than.
18
Another example

• Cobra Cheese Company buys milk from several
  suppliers. Cobra suspects that some producers
  are adding water to their milk to increase their
  profits. Excess water can be detected by
  measuring the freezing point of the milk. The
  freezing temperature of natural milk varies
  normally, with mean µ -0.545C and standard
  deviation s 0.008C. Added water raises the
  freezing temperature toward 0C, the freezing
  point of water. Cobras laboratory manager
  measures the freezing temperature of five
  consecutive lots of milk from one producer. The
  mean measurement is x-bar -0.538C. Is this
  good evidence that the producer is adding water
  to the milk?

19
Homework

• Chapter 9
• 1-4, 14, 16, 17