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?
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 allow me to decide if the
claim is true or not!
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Steps for doing a hypothesis test
Since the p-value lt (gt) a, I reject (fail to
reject) the H0. There is (is not) sufficient
evidence to suggest that Ha (in context).

1. Assumptions
2. Write hypotheses define parameter
3. Calculate the test statistic p-value
4. Write a statement in the context of the problem.

H0 m 12 vs Ha m (lt, gt, or ?) 12
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Assumptions for t-inference

• Have an SRS from population (or randomly assigned
  treatments)
• s unknown
• Normal (or approx. normal) distribution
• Given
• Large sample size
• Check graph of data

Use only one of these methods to check normality
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Formulas

• s unknown

m
t
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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) two-tail test t 2.3 n 25

P-value .0630
P-value (.0152)2 .0304
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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. Is there sufficient evidence that the
bottler is under-filling the bottles?
Use a .1 299.4 297.7 298.9 300.2 297
301
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SRS?

• I have an SRS of bottles

Normal? How do you know?

• Since the boxplot is approximately symmetrical
  with no outliers, the sampling distribution is
  approximately normally distributed

Do you know s?

• s is unknown

What are your hypothesis statements? Is there a
key word?
H0 m 300 where m is the true mean amount Ha m
lt 300 of cola in bottles
p-value .0880
a .1
Plug values into formula.
Compare your p-value to a make decision
Since p-value lt a, I reject the null hypothesis.
Write conclusion in context in terms of Ha.
There is sufficient evidence to suggest that the
true mean cola in the bottles is less than 300 mL.
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Example 2 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

H0 m 34 where m is the true mean reading Ha m
? 34 ability of the districts third-graders

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.
A type II error We decide that the true mean
reading ability is not different from the
national average when it really is different.
What type of error could you potentially have
made with this decision? State it in context.
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What confidence level should you use so that the
results match this hypothesis test?
90
Compute the interval. What do you notice about
the hypothesized mean?
(32.255, 37.927)
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Example 3 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
lower than 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 lt 1323 per week
• Since p-value lt a of 0.05, I reject the null
  hypothesis. There is sufficient evidence to
  suggest that the sales of cookies are lower than
  the earlier figure.

What is the potential error in context? What is
a consequence of that error?
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• Example 3 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 90
  confidence interval for the mean weekly sales
  rate.
• CI (1122.70, 1293.30)
• Based on this interval, is the mean weekly sales
  rate statistically less than the reported 1323?

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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 be an 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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• Stroop Test
• Is there an interaction between color word?
• Or in other words is there a significant
  increase in time?

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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
Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Morning 8 9 7 9 10 13 10 8 2 5 7 7 6 8 7
After-noon 8 10 9 8 9 11 8 10 4 7 8 9 6 6 9
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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Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Morning 8 9 7 9 10 13 10 8 2 5 7 7 6 8 7
After-noon 8 10 9 8 9 11 8 10 4 7 8 9 6 6 9
Differences 0 -1 -2 1 1 2 2 -2 -2 -2 -1 -2 0 2 -2
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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Differences 0 -1 -2 1 1 2 2 -2 -2 -2 -1 -2 0 2 -2
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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Differences 0 -1 -2 1 1 2 2 -2 -2 -2 -1 -2 0 2 -2
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
How could I increase the power of this test?