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Significance Tests
Section 10.2.1
Cookie Monster’s Starter
• Me like Cookies! Do you?
• You choose a card from my deck.
• If card is red, I give you coupon for one
cookie at the cafeteria!
• Let’s play!
Today’s Objectives
• Form a null hypothesis and an alternative
hypothesis about a population parameter.
• Find the P-value in support of the alternative
hypothesis.
• Write a conclusion about the evidence in a
three-phrase form.
California Standard 18.0
Students determine the P- value for a statistic for a
simple random sample from a normal distribution.
The Reasoning of a Significance Test
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Confidence intervals are used to estimate the
value of some population parameter.
Significance tests are used to support some
claim about a parameter.
The logic involves these four steps:
1. State what is known about the parameter.
a) What value do we ASSUME true?
b) What value is CLAIMED true?
2. Gather evidence about the population (such as a
sample mean or sample proportion).
3. Ask whether the results found could have happened
by chance alone if the assumption (1a) is true.
4. Draw a conclusion about the claim (1b).
Step 1: State What is Assumed and Claimed
• State the assumption about the population
that is held before we gather evidence.
– This is called the null hypothesis.
– The notation used is Ho
• State the claim that is to be proven.
– This is called the alternative hypothesis.
– The notation used is Ha
• Write Ho and Ha as a pair of inequalities.
– We normally form Ha first because it is easiest
to understand.
– When actually written on paper, Ho comes first.
Example
• In a criminal trial, the defendant is
assumed innocent until proven guilty.
• The assumption (that he is innocent) is the
null hypothesis.
• The alternative hypothesis is that he is
guilty and must be supported by evidence.
• So the notation would read:
Ho: Defendant is innocent.
Ha: Defendant is guilty.
• Note that we never prove innocence.
– We just decide whether there is sufficient
evidence to support Ha
Step 2: Gather evidence about the
population
• Determine methodology and sample size
• Collect data
• Calculate sample mean or sample
proportion
– This is known as the test statistic
Step 3: Ask whether the evidence
gathered could have happened by
chance alone
• Based on the assumption in Ho, find the
sampling distribution of the statistic.
– What are the mean and standard deviation for x-bar
or p-hat?
• Find the probability of getting a statistic as
extreme (or more extreme) as the one gathered.
– Notice that this probability is based on the assumption
that Ho is true.
– The probability is called the P-value.
Step 4: Draw a conclusion about
the claim
• Is the probability low enough that you
believe the statistic gathered could not
have happened by chance?
– Generally we tend to believe that a 5% or
10% (or higher) event could occur by chance.
– If the probability is less than 5%, we tend to
get skeptical.
• (Remember the cards)
• If the probability is too low, take that as
evidence in support of Ha
Example
• I used to own a Baskin-Robbins ice cream store
in Pleasant Hill (really!).
• I trained my employees to make their scoops
weigh 3.5 oz.
– Specifically, scoops are N(3.5 oz, 0.1 oz)
• Two of my first employees were Gina Z and Jim
Wrenn.
– (Yes, the Mr. Wrenn who taught at NHS).
• My wife thought that Gina and Jim were
scooping too much ice cream, so I did a study.
• I weighed 10 scoops randomly from each.
– Jim’s averaged 3.58 oz
– Gina’s averaged 3.52 oz
• Is this evidence to support my wife’s claim, or
could this happen by random chance?
Form the hypotheses
• We are trying to prove that the scoops
weigh more than 3.5 oz, so that is the
alternative hypothesis.
• We assume that the scoops are 3.5 oz (or
less), so that is the null hypothesis.
• Here is the notation:
Ho: µ = 3.5 oz
Ha: µ > 3.5 oz
Gather evidence and find P-value for Jim
• Jim’s sample mean was 3.58
• Distribution of sample means is N(3.5, .03)
– We are assuming Ho true
– Standard deviation was found by σ/√n formula
• What is the probability of getting a result as
high as 3.58 or higher under this distribution?
– normalcdf(3.58, 999, 3.5, .03) = .004
– Note that this is the P-value
Draw a conclusion
• This says that if Jim’s usual scooping is
really N(3.5, .1), there is about a 0.4%
probability of getting a sample mean this
high (or higher) by chance alone.
• I don’t believe that an event with such low
probability could have happened by
chance, so I take this as evidence that the
mean of his distribution is really higher
than 3.5.
– In other words, this is strong evidence to
support Ha
Write the conclusion
• You must write your conclusion as a full
sentence in context.
• Use a three-phrase model:
– Because the P-value of ### is so (low/high)…
– there (is/is not) good evidence to support the claim
that…
– Re-state the claim. (IN CONTEXT!!!)
• Follow this model to write a conclusion to this
problem now.
• Because the P-value of 0.4% is so low, there is
good evidence to support the claim that Jim’s
scoops average more than 3.5 oz.
Do the analysis for Gina
• Calculate her P-value (based on  = 3.52).
• Decide whether the result could have happened by
chance.
• Write a three-phrase conclusion.
• P-value= normalcdf(3.52, 999, 3.5, .03) = .252
• A 25% probability could easily happen by chance.
• Because the P-value of .252 is so high, there is not
good evidence to support the claim that Gina’s
scoops average more than 3.5 oz.
Three forms of hypotheses
• If the claim is that the population mean is
greater than some value k:
– Ho: µ=k
Ha: µ>k
• If the claim is that the population mean is
less than some value k:
– Ho: µ=k
Ha: µ<k
• If the claim is that the population mean is
different than some value k:
– Ho: µ=k
Ha: µ≠k
• Note that you must choose Ho and Ha
before gathering data
Today’s Objectives
• Form a null hypothesis and an alternative
hypothesis about a population parameter.
• Find the P-value in support of the alternative
hypothesis.
• Write a conclusion about the evidence in a
three-phrase form .
California Standard 18.0
Students determine the P- value for a statistic for a
simple random sample from a normal distribution.
Homework
• Read pages 531 - 539
• Do problems 27 - 31