APSTAT UNIT 4A INFERENCE PART 1

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APSTAT UNIT 4AINFERENCE PART 1
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APSTAT Chapter 18Sampling Distributionsand
Sample Means
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Lets Just DO IT!!!!
Proportion of correct answers on last AP Stat Exam
Regular Old Distribution
.55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .8
5-.89
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Lets Just DO IT!!!!
Proportion of correct answers on last AP Stat Exam

• Now, Everyone take a 5-person random sample
• Do randint(1,13,5) to choose your subjects
• Add their scores and divide by 5 to get x-bar
  (sample mean)
• Now we will do a distribution of our sample
  means a SAMPLING DISTRIBUTION!!!!!

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Lets Just DO IT!!!!
Proportion of correct answers on last AP Stat Exam
Class Sample Means
Sampling Distribution
.55-.59 .60-.64 .65-.69 .70-.74 .75-.79 .80-.84 .8
5-.89
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We Just DID IT!!!!

• Give me at least 2 things that are different
  between the regular distribution and the sampling
  distributions

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Bias

• Unbiased Statistic
• Mean of Sampling distribution should equal True
  population mean.
• How did ours look earlier? The true mean of the
  population was about 72.3.

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Sample Proportions

• Mean of Sample Proportion
• In last section
• Proportion is just outcome divided by n, so.

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Standard Deviation of Sample Proportion

• In last section
• Proportion is just outcome divided by n, so throw
  down a little Algebra.

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Now try itSample Proportions

• Find mean and standard deviation for
• 60 samples of 10 coin flips, p.5
• 60 samples of 50 coin flips, p.5
• What does this say about variability in regards
  to sample size?

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2 Rules of Thumb - Assumptions/Conditions

• Population size large enough
• Population should be at least 10 times the sample
  size
• 10 Condition
• Normalness
• n should be large enough to produce an
  approximately normal sampling distribution.
• np gt 10 AND n(1-p) gt10

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Try them out

• A San Jose firm decide to sample 25 residents to
  determine if they oppose off-shore oil drilling.
  They predict that P(oppose) 0.4
• Large enough population?
• Normalness?

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Example.

• If the true percentage of students who pass the
  APStat exam is .64, what is the probability that
  a random sample of 100 students will have at
  least 70 students pass?

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?.64, n100, 70 or more

• Check Conditions - Briefly Explain
• 10
• np and n(1-p) gt 10
• Draw Picture- (Find SD too)
• Find P-Value
• Conclusion

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Same Problem Data

• Within what range would we expect to find 95 of
  sample proportions of size 100.

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Sample Means

• Take a whole bunch of samples and find the means
• Why sample means?
• Remember our sample of class scores?
• Less variable
• More normal

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Mean and Standard Deviation of X-BAR

• If we take all the possible samples from a
  population, the mean of the sampling distribution
  will equal the population mean (if the population
  mean was accurate in the first place, but more on
  that later)

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Mean and Standard Deviation of X-BAR

• Standard Deviation of a sampling distribution is

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Lets try it!

• If adult males have height N(68,2) what would be
  the mean and standard deviation for the
  distribution if
• n10
• n40
• What happened to the Standard deviation when n
  was quadrupled?
• What would happen to the standard deviation if n
  was multiplied by 9?

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CLT The Central Limit Theorem

• If the population we are sampling from is already
  normal with N(?,?), the sampling distribution
  will be normal as well with mean ? and standard
  deviation
• But what if the population we are sampling from
  is not normal?

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Age of Pennies

• Riebhoff has 50 pennies, he took the current year
  and subtracted it from the date on the penny to
  obtain the following data

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Penny Ages
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Sample Size n1
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Sample Size n4(everyone do 3 SRS)
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Sample Size n8(everyone do 3 SRS)
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What happened?

• The distribution got normaler as the sample
  size increased. Cool?
• Central Limit Theorem says that even if a
  distribution is not normal, the distribution of
  the sampling distribution will approach normalcy
  when n is large.
• Allows us to use z-scores and such, even when the
  larger population is not normally distributed.

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Assumptions/Conditions

• Random Sample - Always describe
• Independence - Describe
• 10 Condition

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Try it

• If the APSTAT EXAM 2005 had a mean score of 3.2
  with a standard deviation of 1.2
• Old Skool - Find the probability that a single
  student would have a score of 4 or higher?
• New Skool find the probability that an SRS of
  20 students would have a score of 4 or higher?

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Old Skool - Find the probability that a single
student would have a score of 4 or higher?
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New Skool find the probability that an SRS of
20 students would have a score of 4 or higher?

• Check Conditions - Briefly Explain
• 10
• Independence and Random Sample
• Draw Picture- (Find SD too)
• Find P-Value
• Conclusion

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Standard Error

• Sometimes we do not have the population standard
  deviation. If we have to estimate it, we call it
  Standard Error and roll an SE.

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APSTAT Chapter 19Confidence Intervals for
Proportions
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Confidence Interval for a Proportion (aka
One-Proportion Z-Interval)

• At Woodside High, 80 students are surveyed and
  32 of them had tried marijuana.
• How confident am I that the true proportion of WH
  students that have tried marijuana is at or near
  32?
• CONFIDENCE INTERVAL!!!

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The Dealio

• If I do know the population mean
• If I sample, I know the sample mean might be
  quite different than the population mean
• BUTThat difference is predictable.
• For instance, if N(0.70,0.1) and n4
• Sample Mean 0.70, Sample SD0.1/sqrt4.05
• We expect 95 of samples (Empirical Rule) to fall
  between 2 SD of the mean
• Therefore 95 of samples will fall between 0.6
  and 0.8.

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Confidence Intervals

• Work in reverse
• (From Woodside High Example) I sampled 80 and got
  sample p .32
• I want to know the true population proportion.
• The true population proportion will lie within
  2SD of the Sample Proportion in 95/100 samples of
  this size.
• Lets Do It!!!!

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Do It!

• List what you know
• p-hat.32, n80
• Conditions/Assumptions
• 10 for Independence
• Woodside HS has over 800 students
• np and nq gt 10 to use Normal Model
• Both .32 x 80 and .68 x 80 gt 10
• Find Standard Error
• SE(p-hat)

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Do It!

• Draw the Picture
• Conclusion
• We are 95 confident that the TRUE mean
  proportion of ____________ falls between ____ and
  ____

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Do It!

• We can also write confidence intervals in the
  form
• (estimate) (margin of error)

Standard error
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What Does 95 Confidence Mean?

• If we did a whole bunch of confidence intervals
  at this sample size, we would expect 95 out of
  100 intervals to contain the true mean.
• Picture of this

TRUE POPULATION PROPORTION
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AHOY!

• We do not always want 95 confidence.
• Example, if a part on an airplanes landing gear
  needs to be a certain size to work, wouldnt you
  want a little more confidence in the sample being
  within certain parameters?
• Common Intervals are 90, 95 and 99
• Denote as C.90, C.95, or C.99

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But 90 and 99 arent Empirically Cool

• We need this z-score! Its critical!
• So critical, it is called the critical value and
  denoted as z

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Mas z

• Now check t distribution critical values chart
  (back of book or formula sheet)
• Look at bottom. It gives you C and right above
  it is..
• Yeah!

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Try it!

• A poll asked who would you vote for if an
  election were held today between Sen. Barack
  Obama and Sen. John McCain. 115 of the 250
  respondents chose Sen. McCain. Construct and
  interpret a 90 confidence interval for the
  proportion of voters choosing McCain.

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Try It!

• Conditions
• Mean, SE, z
• Calculate CI
• Conclusion

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Last thing

• Finding sample size needed for a CI with a given
  level of confidence and a given margin of error
• NBC News is doing a poll on who will be the next
  Governor of California. The want a 3 margin of
  error at a 95 confidence interval. What sample
  size should they use?

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Sample size needed
Margin of Error
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Sample size needed
Why 0.5? Gives us largest n value. Safety First!
OOPS! YOU SHOULD ALWAYS ROUND UP TO STAY WITHIN
CONFIDENCE INTERVAL! SHOULD BE 1068.
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APSTAT Chapter 20One ProportionHypothesis
Tests
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Significance Tests

• Example. AP Stat Exam 2005
• National Proportion Who Passed .58
• Priory Students n 32, p-hat.78
• Two Possibilities
• Higher WPS proportion just happened by chance
  (natural variation of a sample)
• The likelihood of 78 of 32 students passing is
  so remote we must conclude that Priory Students
  are likely better at APStat than national
  average.

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Hypothesis Testing

• Reflect our two possibilities from above
• NOTHING IS STRANGE (difference could have been by
  natural variation of sample)
• SOMETHIN IS GOIN ON (difference is so
  improbable we must assume there is a difference)
• Here is how we write them
• H0 Null Hypothesis (Nothing Strange)
• Ha Alternative Hypothesis (Somethin is goin on)

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In our WPS SAT Example

• In practice, we describe the hypotheses in both
  symbols and words
• H0 p .58, Priory students perform at the same
  level as the National Average
• Ha p gt .58, Priory students perform better than
  the National Average
• We will perform test(s) that give evidence
  against the H0 (kinda like a trial)

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What to do with the Hypothesis

• After we conduct a test we will have evidence
  based on our understanding of probability and
  sample variation. With this info we can
• Reject H0 in favor of Ha
• if there is SIGNIFICANT evidence that the result
  did not likely happen by chance variation.
• Fail to Reject H0
• if there is not enough evidence to reject it.
  The variation could likely have happened by chance

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Be Carefull

• Notice we NEVER, NEVER, NEVER
• Accept either Hypothesis
• Say one or the other is true or false
• We only have evidence, we could still be wrong.
• BUT.the stronger the evidence the more confident
  we can be!

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Where do we get evidence?

• One way, P-value from a z-score. What is the
  probability that this event happened given the
  population mean, standard deviation and in our
  trial?
• Our old friend, the z-score

We are using a sample here, so we throw in our
sample standard deviation.
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Lets Do It! WPS SAT Example

• Step 1 Define Parameter
• p the true passing proportion of WPS APstat
  test-takers
• Step 2 Hypotheses
• H0 p .58, Priory students perform at the same
  level as the national proportion
• Ha pgt .78, Priory students perform better than
  the national proportion

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WPS SAT Example Continued

• Step 3 Assumptions
• SRS
• No, but we will assume WPS Students are a
  representative sample of the population of all AP
  Stat test-takers.
• Independence
• Priory sample of 32 is less than 10 of
  population of AP Stat test-takers
• ????????????????
• .58(32) and .42(32) both gt 10

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WPS SAT Example Continued

• Step 4 Name Test and DO IT
• One Sample Z-Test for a Proportion

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WPS SAT Example Continued

• Step 5 P-value and sketch of normal curve
• P(zgt 2.31) .01053
• Step 6 Interpret P-value and Conclusion
• A P-Value of .01053 indicates that there is about
  a 1 in 100 chance that a result this distant from
  the p happened merely by chance. Therefore,
  reject H0 in favor of Ha. It is very likely that
  WPS students performed far better on average than
  the National Average on the 2005 APStat exam

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PHAT-PI (MUCH LOVE TO AL YOUNG)

• P - Parameters (What are we studying)
• H - Hypothesis (In words and symbols)
• A - Assumptions (depends on type of test)
• T - Test (Name it. Do it.)
• P - P-Value (Calculate it-Draw it)
• I - Interpret (Reject/Fail to Reject, Why, ATQ)

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Alternative Hypotheses

• Can be
• Greater Than (Ha ------gtblah)
• Less Than (Ha --------ltblah)
• Not (Ha --------?blah)
• Double your one-sided P-value

?0
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On TI-83

• Still have to do all of Phat Pi, but helps with
  calculations.
• StatgtTestgt1-PropZTest
• p0 - Population proportion
• x - successes in sample n(p-hat)
• n - sample size
• Do it for AP Stat Example

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Defective Products

• A company claims that just 3 of its products are
  defective. A simple random sample of 400 of
  their products yielded 14 defective items. Do
  these sample data suggest that the companys
  claim is too low?

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PHAT-PI

• P
• H
• A

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PHAT-PI

• T
• P
• I

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How Much Evidence?

• GTang (and many texts) give a rule of thumb of
  5. If there is a 5 probability or less that
  the outcome would happen by chance, you can throw
  down the enough evidence to reject H0
• If it is 1 or less, you can throw down the very
  strong evidence against H0. Reject H0 in favor

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Significance level

• Sometimes a problem will specify a certain amount
  of evidence that is needed.
• ? Significance Level
• Usually ? 0.05 or 0.01
• Basically, your P-value must be below that level
  to reject the null hypothesis.
• Example your p-value is .03 and ? 0.05
• Be careful with one and two-sided alternatives
  and significance levels
• Your p-value doubles in a 2-sided.

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APSTAT Chapter 21More Stuff About Hypothesis
Tests
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Great Chapter

• Make sure you read it
• Important concepts
• What a Null Hypothesis is and isnt
• What P-Value Means
• Significance (?) Level
• Critical Value - One v. Two sided
• Confidence Intervals and Tests of Significance -
  Relationship Between

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Great ChapterBut.

• Goes further than you need in explaining
• Types of Error
• Type I
• Type II
• Power

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Errors - Can We Make Mistakes?

• Sure, Rejecting a Good Shipment
• For Example, I need batteries that work 99 of
  the time. My significance test of a sample from
  a battery shipment tells me to reject the
  shipment, but it is actually ok.
• We Can also Fail to Reject a Bad Statement
• If I had accepted a shipment that was actually
  bad because my sample proportion ended up close
  to the mean I was looking for.
• Which of these is worse in real life?

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Errors

• Type I Reject H0 when it is actually true
• Usually not so bad
• Rejecting a good shipment
• Probability is equal to ?
• Type II Failing to Reject H0 when it is
  actually false
• Usually bad
• Accepting a bad shipment
• Probability (?) is a bear to calculate
• Check book to see how! Ooooo, fun!
• Be happy you will NEVER be asked to do it

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Errors - 2

• Decrease both Type I and II errors by
• Increasing n
• Decrease Type II Errors by
• Increasing ?
• You end up rejecting more/failing to reject less
• Causes an increase in Type I errors

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POWER

• Basically, how sure we are that we will not get a
  Type II error
• Power 1 P(Type II)
• OR Power 1 - P(?)
• Never will you be asked to compute (unless the
  probability of a type II error is given)
• Increase Power by
• Increasing n (Sample size)
• Increase ? (say from .01 to .05)

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Power and Error Wrap

• What you have to know
• Explain Power, Type I, and Type II errors in
  context of the problem.
• Calculate P(Type I error) given ?
• How to Decrease
• Type I Error
• Type II Error
• How to increase Power

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APSTAT Chapter 22Two Proportion Hypothesis
Tests
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Lets Hop Right In

• A recent report found that men wash their hands
  75 of the time after using the restroom and
  women 85 of the time. If SRSs of 1200 men and
  1100 women were surveyed, can we statistically
  say there is a significant difference between
  hand washing habits of men and women?

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Handwashing

• Parameter (group 1female, 2male)
• ?1-p2 Difference between female and male hand
  washing proportions
• Hypotheses
• H0 p1-p20 No difference in hand washing
• Ha p1-p2?0 Is a

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Handwashing

• Assumptions
• SRSs Yep
• Independent samples Safe to Assume
• n1p1gt5 and n1(1-p1)gt5
• n2p2gt5 and n2(1-p2)gt5 Yep
• Population 10X Sample Yep

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Handwashing

• Test Two Sample Proportion Z-Test

POOLED
Pool if variances are equal (since our null
theorizes that the populations and thus the
variances - are equal)
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Handwashing

• P-Value
• 2P(Zgt5.965)Really Really Really Small
• Interpretation
• P Value is so small, there is VERY significant
  evidence against the assumption that males and
  females wash hands at the same proportion.
  Reject Null Hypothesis in favor of the
  Alternative. Males and females almost assuredly
  have different hand washing proportions.

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Pooled vs. Non-Pooled

• Use Pooled when you hypothesize populations have
  the same variance (in proportions, the same p
  same variance)
• Use Non-pooled when populations are likely to
  have separate variances. (If your null shows a
  non-zero difference)

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Confidence Interval
Use Non-Pooled because there is no null to test
for.
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So, To Review

• PhatPi is on, but with these changes
• P Parameter of interest is now the difference
  between ___ and ___
• H H0 p1p2 (or p1-p20)
• Ha p1gtp2 (or p1-p2gt0)
• or Ha p1ltp2 (or p1-p2lt0)
• or Ha p1?p2 (or p1-p2 ? 0)
• Plus, you have to choose Pooled v. Non-Pooled
  (Pooled if Null is p1p2)

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Using TI 83

• StatgtTestgt2-PropZTest
• Can Also Do Interval
• StatgtTestgt2-PropZInt
• Put in C-Level (usually .9, .95, or .99)

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Lets do one!

• Some scientist suggest that sickle-cell traits
  protect against malaria. A study in Africa
  tested 543 for sickle-cell trait and also for
  malaria. In all, 136 of the children had
  sickle-cell trait and 36 of these had malaria.
  The other 407 children lacked the sickle-cell
  trait and 157 of them had malaria. Is there
  evidence that malaria infection is lower among
  children with the sickle-cell trait.

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Malaria v. Sickle Cell

• P
• H
• A

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Malaria v. Sickle Cell

• T
• P
• I

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Do Using a 95 C-Interval

• Assumptions
• Interval Calculation
• Interpretation

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That is it!

• Just one section left to go!