Hypothesis Tests One Sample Means

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Hypothesis Tests One Sample Means
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How can I tell if they really are underweight?
Example A government agency has received
numerous complaints that a particular restaurant
has been selling underweight hamburgers. The
restaurant advertises that its patties are a
quarter pound (4 ounces).
A hypothesis test will help me decide!
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What are hypothesis tests?

• Calculations that tell us if a value, x, occurs
  by random chance or not if it is statistically
  significant
• Is it . . .
• a random occurrence due to natural variation?
• a biased occurrence due to some other reason?

Is it one of the sample means that are likely to
occur?
Statistically significant means that it is NOT a
random chance occurrence!
Is it one that isnt likely to occur?
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Nature of hypothesis tests -
How does a murder trial work?

• First begin by supposing the effect is NOT
  present
• Next, see if data provides evidence against the
  supposition
• Example murder trial

First - assume that the person is innocent Then
must have sufficient evidence to prove guilty
Hmmmmm Hypothesis tests use the same process!
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Steps
Notice the steps are the same except we add
hypothesis statements which you will learn today

• Assumptions
• Hypothesis statements define parameters
• Calculations
• Conclusion, in context

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Assumptions for z-test (t-test)
YEA These are the same assumptions as
confidence intervals!!

• Have an SRS of context
• Distribution is (approximately) normal
• Given
• Large sample size (CLT)
• Graph data
• s is known (unknown)

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• Example 1 Bottles of a popular cola are supposed
  to contain 300 mL of cola. There is some
  variation from bottle to bottle. An inspector,
  who suspects that the bottler is under-filling,
  measures the contents of six randomly selected
  bottles. Are the assumptions met?
• 299.4 297.7 298.9 300.2 297 301

• Have an SRS of bottles
• Sampling distribution is approximately
• normal because the boxplot is
• symmetrical
• s is unknown

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Writing Hypothesis statements

• Null hypothesis is the statement being tested
  this is a statement of no effect or no
  difference
• Alternative hypothesis is the statement that we
  suspect is true

H0
Ha
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The form

• Null hypothesis
• H0 parameter hypothesized value
• Alternative hypothesis
• Ha parameter gt hypothesized value
• Ha parameter lt hypothesized value
• Ha parameter hypothesized value

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Example 2 A government agency has received
numerous complaints that a particular restaurant
has been selling underweight hamburgers. The
restaurant advertises that its patties are a
quarter pound (4 ounces). State the hypotheses
H0 m 4 Ha m lt 4
Where m is the true mean weight of hamburger
patties
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Example 3 A car dealer advertises that is new
subcompact models get 47 mpg. You suspect the
mileage might be overrated. State the hypotheses

H0 m 47 Ha m lt 47
Where m is the true mean mpg (this is not
sufficient add to subcompact models)
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Example 4 Many older homes have electrical
systems that use fuses rather than circuit
breakers. A manufacturer of 40-A fuses wants to
make sure that the mean amperage at which its
fuses burn out is in fact 40. If the mean
amperage is lower than 40, customers will
complain because the fuses require replacement
too often. If the amperage is higher than 40,
the manufacturer might be liable for damage to an
electrical system due to fuse malfunction. State
the hypotheses
H0 m 40 Ha m 40
Where m is the true mean amperage of the fuses
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Facts to remember about hypotheses

• ALWAYS refer to populations (parameters)
• The null hypothesis for the difference between
  populations is usually equal to zero

H0 mx-y 0
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Activity For each pair of hypotheses, indicate
which are not legitimate explain why
Must be NOT equal!
p is the population proportion!
Must use same number as H0!
r is parameter for population correlation
coefficient but H0 MUST be !
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P-values -

• Assuming H0 is true, the probability that the
  test statistic would have a value as extreme or
  more than what is actually observed

In other words . . . is it far out in the tails
of the distribution?
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Level of significance -

• Is the amount of evidence necessary before we
  begin to doubt that the null hypothesis is true
• Is the probability that we will reject the null
  hypothesis, assuming that it is true
• Denoted by a
• Can be any value
• Usual values 0.1, 0.05, 0.01
• Most common is 0.05

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Statistically significant

• The p-value is as small or smaller than the level
  of significance (a)
• If p gt a, fail to reject the null hypothesis at
  the a level.
• If p lt a, reject the null hypothesis at the a
  level.

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Facts about p-values

• ALWAYS make decision about the null hypothesis!
• Large p-values show support for the null
  hypothesis, but never that it is true!
• Small p-values show support that the null is not
  true.
• Double the p-value for two-tail () tests
• Never accept the null hypothesis!

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Never accept the null hypothesis!
Never accept the null hypothesis!
Never accept the null hypothesis!
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At an a level of .05, would you reject or fail to
reject H0 for the given p-values?

• .03
• .15
• .45
• .023

Reject
Fail to reject
Fail to reject
Reject
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Calculating p-values

• For z-test statistic
• Use normalcdf(lb,ub)
• using standard normal curve
• For t-test statistic
• Use tcdf(lb, ub, df)

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Draw shade a curve calculate the p-value

• 1) right-tail test t 1.6 n 20
• 2) left-tail test z -2.4 n 15
• 3) two-tail test t 2.3 n 25

P-value .0630
P-value .0082
P-value (.0152)2 .0304
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Writing Conclusions

• A statement of the decision being made (reject or
  fail to reject H0) why (linkage)
• A statement of the results in context. (state in
  terms of Ha)

AND
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• Since the p-value lt (gt) a, I reject (fail to
  reject) the H0. There is (is not) sufficient
  evidence to suggest that Ha.

Be sure to write Ha in context (words)!
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• Example 5 Drinking water is considered unsafe if
  the mean concentration of lead is greater than 15
  ppb (parts per billion). Suppose a community
  randomly selects of 25 water samples and computes
  a t-test statistic of 2.1. Assume that lead
  concentrations are normally distributed. Write
  the hypotheses, calculate the p-value write the
  appropriate conclusion for a 0.05.

H0 m 15 Ha m gt 15 Where m is the true mean
concentration of lead in drinking water
Since the p-value lt a, I reject H0. There is
sufficient evidence to suggest that the mean
concentration of lead in drinking water is
greater than 15 ppb.
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• Example 6 A certain type of frozen dinners
  states that the dinner contains 240 calories. A
  random sample of 12 of these frozen dinners was
  selected from production to see if the caloric
  content was greater than stated on the box. The
  t-test statistic was calculated to be 1.9. Assume
  calories vary normally. Write the hypotheses,
  calculate the p-value write the appropriate
  conclusion for a 0.05.

H0 m 240 Ha m gt 240 Where m is the true mean
caloric content of the frozen dinners
Since the p-value lt a, I reject H0. There is
sufficient evidence to suggest that the true mean
caloric content of these frozen dinners is
greater than 240 calories.
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Formulas

• s known

m
z
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Formulas

• s unknown

m
t
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Example 7 The Fritzi Cheese Company buys milk
from several suppliers as the essential raw
material for its cheese. Fritzi suspects that
some producers are adding water to their milk to
increase their profits. Excess water can be
detected by determining the freezing point of
milk. The freezing temperature of natural milk
varies normally, with a mean of -0.545 degrees
and a standard deviation of 0.008. Added water
raises the freezing temperature toward 0 degrees,
the freezing point of water (in Celsius). The
laboratory manager measures the freezing
temperature of five randomly selected lots of
milk from one producer with a mean of -0.538
degrees. Is there sufficient evidence to suggest
that this producer is adding water to his milk?
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SRS?
Assumptions
Normal? How do you know?

• I have an SRS of milk from one producer

• The freezing temperature of milk is a normal
  distribution. (given)

Do you know s?

• s is known

What are your hypothesis statements? Is there a
key word?
H0 m -0.545 Ha m gt -0.545 where m is the
true mean freezing temperature of milk
Plug values into formula.
p-value normalcdf(1.9566,1E99).0252
Use normalcdf to calculate p-value.
a .05
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Compare your p-value to a make decision
Conclusion
Since p-value lt a, I reject the null hypothesis.
There is sufficient evidence to suggest that the
true mean freezing temperature is greater than
-0.545. This suggests that the producer is
adding water to the milk.
Write conclusion in context in terms of Ha.
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Example 8 The Degree of Reading Power (DRP) is a
test of the reading ability of children. Here
are DRP scores for a random sample of 44
third-grade students in a suburban
district (data on note page) At the a .1, is
there sufficient evidence to suggest that this
districts third graders reading ability is
different than the national mean of 34?
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SRS?

• I have an SRS of third-graders

Normal? How do you know?

• Since the sample size is large, the sampling
  distribution is approximately normally
  distributed
• OR
• Since the histogram is unimodal with no outliers,
  the sampling distribution is approximately
  normally distributed

Do you know s?
What are your hypothesis statements? Is there a
key word?

• s is unknown

Plug values into formula.
p-value tcdf(.6467,1E99,43).2606(2).5212
Use tcdf to calculate p-value.
a .1
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Compare your p-value to a make decision
Conclusion
Since p-value gt a, I fail to reject the null
hypothesis.
There is not sufficient evidence to suggest that
the true mean reading ability of the districts
third-graders is different than the national mean
of 34.
Write conclusion in context in terms of Ha.
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Example 9 The Wall Street Journal (January 27,
1994) reported that based on sales in a chain of
Midwestern grocery stores, Presidents Choice
Chocolate Chip Cookies were selling at a mean
rate of 1323 per week. Suppose a random sample
of 30 weeks in 1995 in the same stores showed
that the cookies were selling at the average rate
of 1208 with standard deviation of 275. Does
this indicate that the sales of the cookies is
different from the earlier figure?
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• Assume
• Have an SRS of weeks
• Distribution of sales is approximately normal due
  to large sample size
• s unknown
• H0 m 1323 where m is the true mean cookie
  sales
• Ha m ? 1323 per week
• Since p-value lt a of 0.05, I reject the null
  hypothesis. There is sufficient to suggest that
  the sales of cookies are different from the
  earlier figure.

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• Example 9 Presidents Choice Chocolate Chip
  Cookies were selling at a mean rate of 1323 per
  week. Suppose a random sample of 30 weeks in
  1995 in the same stores showed that the cookies
  were selling at the average rate of 1208 with
  standard deviation of 275. Compute a 95
  confidence interval for the mean weekly sales
  rate.
• CI (1105.30, 1310.70)
• Based on this interval, is the mean weekly sales
  rate statistically different from the reported
  1323?

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What do you notice about the decision from the
confidence interval the hypothesis test?
A 96 CI (1100, 1316). Since 1323 is not in
the interval, we would reject H0.

• What decision would you make on Example 9 if a
  .01?
• What confidence level would be correct to use?
• Does that confidence interval provide the same
  decision?
• If Ha m lt 1323, what decision would the
  hypothesis test give at a .02?
• Now, what confidence level is appropriate for
  this alternative hypothesis?

You would fail to reject H0 since the p-value gt a.
Remember your, p-value .01475 At a .02, we
would reject H0.
You should use a 99 confidence level for a
two-sided hypothesis test at a .01.
The 98 CI (1084.40, 1331.60) - Since 1323
is in the interval, we would fail to reject
H0. Why are we getting different answers?
Tail probabilities between the significant level
(a) and the confidence level MUST match!)
In a CI, the tails have equal area so there
should also be 2 in the upper tail
CI (1068.6 , 1346.40) - Since 1323 is in
this interval we would fail to reject H0.
a .02
.02
.96
That leaves 96 in the middle that should be
your confidence level
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Matched Pairs Test

• A special type of
• t-inference

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Matched Pairs two forms

• Pair individuals by certain characteristics
• Randomly select treatment for individual A
• Individual B is assigned to other treatment
• Assignment of B is dependent on assignment of A

• Individual persons or items receive both
  treatments
• Order of treatments are randomly assigned or
  before after measurements are taken
• The two measures are dependent on the individual

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Is this an example of matched pairs?

• 1)A college wants to see if theres a difference
  in time it took last years class to find a
  job after graduation and the time it took the
  class from five years ago to find work after
  graduation. Researchers take a random sample
  from both classes and measure the number of days
  between graduation and first day of employment

No, there is no pairing of individuals, you have
two independent samples
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Is this an example of matched pairs?

• 2) In a taste test, a researcher asks people in a
  random sample to taste a certain brand of spring
  water and rate it. Another random sample of
  people is asked to taste a different brand
  of water and rate it. The researcher wants to
  compare these samples

No, there is no pairing of individuals, you have
two independent samples If you would have the
same people taste both brands in random order,
then it would bean example of matched pairs.
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Is this an example of matched pairs?

• 3) A pharmaceutical company wants to test its new
  weight-loss drug. Before giving the drug to a
  random sample, company researchers take a weight
  measurement on each person. After a month
  of using the drug, each persons weight is
  measured again.

Yes, you have two measurements that are dependent
on each individual.
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A whale-watching company noticed that many
customers wanted to know whether it was better to
book an excursion in the morning or the
afternoon. To test this question, the company
collected the following data on 15 randomly
selected days over the past month. (Note
days were not consecutive.)
You may subtract either way just be careful
when writing Ha
Since you have two values for each day, they are
dependent on the day making this data matched
pairs
First, you must find the differences for each day.
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I subtracted Morning afternoon You could
subtract the other way!

• Assumptions
• Have an SRS of days for whale-watching
• s unknown
• Since the normal probability plot is
  approximately linear, the distribution of
  difference is approximately normal.

You need to state assumptions using the
differences!
Notice the granularity in this plot, it is still
displays a nice linear relationship!
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Is there sufficient evidence that more whales are
sighted in the afternoon?
Be careful writing your Ha! Think about how you
subtracted M-A If afternoon is more should the
differences be or -? Dont look at numbers!!!!
If you subtract afternoon morning then Ha mDgt0
H0 mD 0 Ha mD lt 0 Where mD is the true mean
difference in whale sightings from morning minus
afternoon
Notice we used mD for differences it equals 0
since the null should be that there is NO
difference.
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finishing the hypothesis test Since p-value
gt a, I fail to reject H0. There is insufficient
evidence to suggest that more whales are sighted
in the afternoon than in the morning.
In your calculator, perform a t-test using the
differences (L3)
Notice that if you subtracted A-M, then your test
statistic t .945, but p-value would be the
same