Transcript Statistics 10.2

AP Statistics
February 2014
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Coin Flipping Example

On a scrap paper record the results of
my coin flips.
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Why did you doubt my truthfulness?
Because the outcome of the coin flipping
experiment is very unlikely.
 How unlikely?

 (.5)^k, where k is the number of flips before
you yelled.
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Supposition (a fancy way of saying
“unsupported”)
Built into the argument that “Mrs. White
is pulling our collective leg” is a
supposition
 What is that supposition?

 “We suppose that the coin is fair.”

Where does the supposition show up?
 .5^k
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The Test of Significance

The test of significance asks the
question:
 “Does the statistic result from a real
difference from the supposition”
 or
 Does the statistic result from just chance
variation?”
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Example:
I claim that I make 80% of my free throw
shots.
 To test my claim, you ask me to take 20 free
throw shots.
 I make only 8 out of 20
 You respond, “I don’t believe your claim. It is
unlikely that an 80% shooter would make
only 8 out of 20.

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Significance Test Procedure

STEP 1: Define the population and parameter
of interest. State the null and alternative
hypotheses in words and symbols.
Population: My free throw shots
 Parameter of interest: Proportion of shots made
 Suppose I am an 80% free throw shooter. This is a
hypothesis and we think it is false. We will call it the null
hypothesis and use the symbol H0 (pronounced H - nought).
H0 : p = 0.8
 You are trying to show that I am worse than an 80% shooter.
Your alternate hypothesis is: Ha : p < 80%

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Significance Test Procedure

STEP 2: Choose the appropriate inference
procedure. Verify the conditions for using
the selected procedure.

We are going to use the Binomial Distribution
Each trial has either a success or a failure
There is a set number of trials
Trials are independent
The probability of a success is constant




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Significance Test Procedure

STEP 3: Calculate the P-value. The Pvalue is the probability that our sample
statistic is that extreme assuming that
H0 is true.

Look at Ha to calculate “What is the probability of
making 8 or fewer shots out of 20?”
 Binomcdf ( 20, .8, 8) = .000102
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Significance Test Procedure

STEP 4: Interpret the results in the context
of the problem.

You reject H0 because the probability of being an 80%
shooter and making only 8 out of 20 shots is extremely
low. You conclude that Ha is correct; the true proportion
is < 80%

There are only two possibilities at this step:
You reject H0 because the probability is so low. We
accept Ha
 You fail to reject H0 because the probability is not low
enough

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Significance Test Procedure
1.
2.
3.
Identify the population of interest and the
parameter you want to draw conclusions about.
State null and alternate hypotheses.
Choose the appropriate procedure. Verify the
conditions for using the selected procedure.
If the conditions are met, carry out the inference
procedure.


4.
Calculate the test statistic.
Find the P-value
Interpret your results in the context of the
problem
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Example
Diet colas use artificial sweeteners to avoid sugar.
These sweeteners gradually lose their sweetness
over time. Manufacturers therefore test new colas
for loss of sweetness before marketing them.
Trained tasters sip the cola along with drinks of
standard sweetness and score the cola on a
“sweetness score” of 1 to 10. The cola is then
stored for a month at high temperature to imitate
the effect of four months’ storage. Each taster
scores the cola again after storage.
 What kind of experiment is this?

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Example
Here’s the data:
 2.0, .4, .7, 2.0, -.4, 2.2, -1.3, 1.2, 1.1, 2.3
 Positive scores indicate a loss of
sweetness.
 Are these data good evidence that the
cola lost sweetness in storage?

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Significance Test Procedure

Step 1: Define the population and
parameter of interest. State null and
alternative hypotheses in words and
symbols.
 Population: Diet cola.
 Parameter of interest: mean sweetness loss.
 Suppose there is no sweetness loss (Nothing
special going on). H0: µ=0.
 You are trying to find if there was sweetness
loss. Your alternate hypothesis is: Ha: µ>0.
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Significance Test Procedure

Step 2: Choose the appropriate inference
procedure. Verify the conditions for using the
selected procedure.
 We are going to use sample mean distribution:
 Do the samples come from an SRS?
○ We don’t know.
 Is the population at least ten times the sample size?
○ Yes.
 Is the population normally distributed or is the sample size
at least 30.
○ We don’t know if the population is normally distributed, and
the sample is not big enough for CLT to come into play.
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Significance Test Procedure

Step 3: Calculate the test static and the Pvalue. The P-value is the probability that
our sample statistics is that extreme
assuming that H0 is true.
 µ=0, x-bar=1.02, σ=1
 Look at Ha to calculate “What is the probability
of having a sample mean greater than 1.02?”
 z=(1.02-0)/(1/root(10))=3.226,
 P(Z>3.226) =.000619=normalcdf(3.226,1E99)
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Significance Test Procedure

Step 4: Interpret the results in the
context of the problem.
 You reject H0 because the probability of
having a sample mean of 1.02 is very small.
We therefore accept the alternate
hypothesis; we think the colas lost
sweetness.
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Limits of z-Test

There is a major limitation to the z-Test,
and that is it can be used only if
both μ and σ are known, which is
unlikely.
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Limits of z-Test

In cases where the population mean is
known, but the population standard
deviation is not, if a sample is sufficiently
large, it is acceptable to use
the estimated population standard
deviation in place of the true population
standard deviation to calculate an
estimated standard error of the mean.
However, there is no 'real' criterion for
how large a sample must be before it is
“sufficiently” large.
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Assignment

Exercises 10.27-10.40 all.
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