Confidence Intervals with Means

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Confidence Intervals with Means
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What is the purpose of a confidence interval?

• To estimate an unknown population parameter

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Formula
Standard deviation of statistic
Critical value
statistic
Margin of error
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• In a randomized comparative experiment on the
  effects of calcium on blood pressure, researchers
  divided 54 healthy, white males at random into
  two groups, taking calcium or placebo. The paper
  reports a mean seated systolic blood pressure of
  114.9 with standard deviation of 9.3 for the
  placebo group. Assume systolic blood pressure is
  normally distributed.
• Can you find a z-interval for this problem? Why
  or why not?

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Students t- distribution

• Continuous distribution
• Unimodal, symmetrical, bell-shaped density curve
• Above the horizontal axis
• Area under the curve equals 1
• Based on degrees of freedom
• df n - 1

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Formula
Standard deviation of statistic
Standard error when you substitute s for s.
Critical value
statistic
Margin of error
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How to find t
Can also use invT on the calculator! Need upper
t value with 5 is above so 95 is
below invT(p,df)

• Use Table B for t distributions
• Look up confidence level at bottom df on the
  sides
• df n 1
• Find these t
• 90 confidence when n 5
• 95 confidence when n 15

t 2.132
t 2.145
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Steps for doing a confidence interval

1. Assumptions
2. Calculate the interval
3. Write a statement about the interval in the
   context of the problem.

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Statement (memorize!!)

• We are ________ confident that the true mean
  context is between ______ and ______.

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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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• Ex. 1) Find a 95 confidence interval for the
  true mean systolic blood pressure of the placebo
  group.

• Assumptions
• Have randomly assigned males to treatment
• Systolic blood pressure is normally distributed
  (given).
• s is unknown
• We are 95 confident that the true mean systolic
  blood pressure is between 111.22 and 118.58.

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Find a sample size

• If a certain margin of error is wanted, then to
  find the sample size necessary for that margin of
  error use

Always round up to the nearest person!
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Ex 4) The heights of PWSH male students is
normally distributed with s 2.5 inches. How
large a sample is necessary to be accurate within
.75 inches with a 95 confidence interval?
n 43
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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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Formulas

• s unknown

m
t
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Calculating p-values

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