Estimating with Confidence!

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do first
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calculator
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Estimating with Confidence!

• date 1/30
• hw 10.1, 10.2, 10.3

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hw 10.1, 10.2, 10.3
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hw 10.1, 10.2, 10.3
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the Relay problem
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the CLT
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statistical confidence

• 25 NHS students were randomly interviewed about
  their sleep patterns. Their mean sleep time was 6
  hours. From this SRS of n25, what can be said
  about the mean sleep time u of NHS students
  (population)
• the sample mean __ has a ___ distr.
• its mean is __
• the s.d. of ___ is ____ ____

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GRAPH the sampling distribution of mean sleep
time of SRS (n25)

• suppose the pop s.d. is 1 hr.

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the sampling distribution of mean sleep time of
SRS (n25)

• the 68-95-99.7 rule says that in about __ of all
  samples, __ will be within __ s.d. of ___ ___ (u)
• in 95 of all samples, the unknown u lies between
  _____ and _____.
• since our sample mean __ 6, we say we are 95
  confident that the unknown NHS mean sleep time
  lies between ___ and ___.

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95 confidence interval for u

• We got these numbers by a method that gives
  correct results 95 of the time.
• form estimate margin of error
• estimate (critical value)(standard error of the
  estimate).

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applet
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calculator activity
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6.2
6.3
6.4
6.5
6.6
5.8
5.7
5.6
5.5
5.4
6.7
6.1
5.9

• sample 25 from N(6,1) pop randNorm(6,1,25)
• find sample mean (stat calc 1)
• construct 95 interval for u
• construct nine more confidence intervals and use
  Zinterval (Stat-Tests)
• how many contain the true pop mean?
• how many would you expect?

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Confidence Intervals

• date 1/31
• hw 10.5, 10.6, 10.7

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homework

• 10.3
• 95 CI for the mean male score
• 95 of all men have scores between __ and __.

population
sampling distribution
95 of all samples give an interval that
contains the pop mean u
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confidence intervals

• A confidence interval has 2 parts
• an interval
• a confidence level, C
• a 95 confidence level C .95

C
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A 95 confidence interval
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confidence levels
Confidence Level Tail Area
90
95
99
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TABLE C
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Critical values

• is the upper p critical value

probability p
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confidence interval for a population mean
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example NAILS

• take SRS size 5 from Pop (class)
• expect nail length to be ____
• plot the data
• assume s.d. of population is __mm
• give a 90 CI for the mean nail length
• would your result differ is nails came from only
  women?

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larger samples give shorter intervals
n 20
n 5
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quiz 10.1B
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LAB CIs
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data semester grades
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LAB CIs

• http//www.cvgs.k12.va.us/DIGSTATS/main/descriptv/
  d_confidence.htm

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• Suppose we are in an orchard which has twenty
  apple trees, and we place ten buckets underneath
  each tree in order to catch falling apples. Let's
  consider what happens to one tree.From the ten
  buckets, we find a sample mean (X-bar total
  number of apples/ten buckets) of 6 apples per
  bucket, with a standard deviation (s) of 2.8
  apples. If we want to be 95 confident that the
  mean number of apples per bucket in the orchard
  is within a range based upon our one tree,
  assuming the collection of apples in each bucket
  is normally distributed,

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we get a confidence interval of

• So based upon our sample and calculations, we are
  95 confident that the mean number of apples per
  bucket for all the buckets in the orchard is
  between ___ and ___.

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Suppose we used the same level of confidence for
all calculations and calculated a confidence
interval for each tree in the orchard.
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LAB How CIs behave

• date 2/1
• hw 10.9, 10.12,
• 10.13, 10.14

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semester grades

• data set (semester 1 grades)
• plot the population
• pop mean ___ pop s.d.___
• describe the shape
• sample (SRS) 4 scores
• write the sample mean ___ ___
• write the confidence level ___ ____
• write the critical value _______
• write the standard error SE ____
• write the margin of error, ME ____
• write the 90 CI for the true pop mean is
• ___ _____ or ( , )
• when you increase the confidence level, the ME
  ______, so the CI gets ______
• when you increase the sample size (try n16), the
  ME ____, so the CI gets _____

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penny activity what happened
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homework p519 10.6
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Confidence intervals

• you choose the confidence level
• we want high confidence and small margin of error
  
• high confidence
• small margin of error

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the margin of error gets smaller when

• gets smaller
• gets smaller
• n gets larger

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• Confidence intervals get wider as the confidence
  increases
• Confidence intervals get narrower as sample size
  increases

C.90
n5
C.95
n20
C.99
n80
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applet confidence intervals
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• I am 99 confident that the true mean is between
  ___ and ___.

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video CI
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• A guy notices a bunch of targets scattered over a
  barn wall, and in the center of each, in the
  "bulls-eye," is a bullet hole. "Wow," he says to
  the farmer, "that's pretty good shooting. How'd
  you do it?" "Oh," says the farmer, "it was easy.
  I painted the targets after I shot the holes."

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• Confidence intervals are a little like that.
  After we make a point estimate (a bullet hole),
  we are going to draw a target (an interval)
  around the point and state the probability that
  real objective is in the target area. The wider
  the target, the greater the probability, as you'd
  expect. Also, the more accurate the shooting, the
  greater the probability.
• To avoid misleading you, we should re-phrase the
  joke. Suppose there's only one target painted on
  the barn, and all of the bullet holes are within
  it, even though they're not all in the center.
  It's still pretty good shooting, and the better
  the shooting, the smaller the target needs to be
  to encompass them all.
• For the reasons we discussed in the last section,
  the accuracy of the shooting depends on how many
  samples we took and on how variable they are
• If they're all over the place, there's no reason
  to be very confident in their mean.
• If they're concentrated in one area, there's good
  reason to think that the real objective is in
  that area.

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activity
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6.2
6.3
6.4
6.5
6.6
5.8
5.7
5.6
5.5
5.4
6.7
6.1
5.9
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6.2
6.3
6.4
6.5
6.6
5.8
5.7
5.6
5.5
5.4
6.7
6.1
5.9
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6.2
6.3
6.4
6.5
6.6
5.8
5.7
5.6
5.5
5.4
6.7
6.1
5.9
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choosing the sample size

• wanted high confidence - small margin of error
• question how large a sample do I need to be
  within ___ of the true mean with ___ confidence?

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some algebra.

• A Subaru dealer wants to find out the age of
  their customers  (for advertising purposes).
   They want the margin of error to be 3 years old.
   If they want a 90 confidence interval, how many
  people do they need to know about?
• Hence the dealer should survey at least 52
  people.

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some cautions

• date 2-2
• hw p526 10.16-10.18

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estimating a normal mean

• data from an SRS from pop
• outliers affect CI (sample mean is sensitive)
• if n is small and pop is not normal confidence
  level C is different.
• the pop s.d. is known (or sample is large)
• undercoverage, nonresponse may be present

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interpreting CI

• we are 95 confident that the true mean ___ lies
  between ___ and ___
• 95 of the population lies between __ and __
• the method gives correct results in 95 of all
  possible samples.
• the probability is 95 that the true mean falls
  between ___ and ___

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tests of significance
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• CI estimate u
• significance test check a claim about u

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• A coach says the average weight an NHS student
  can lift is of 30 kilograms. You took a sample of
  100 NHS students. The average weight lifted of
  this sample was 28 kilos, with a standard
  deviation of 12 kilos. Does this enable you to
  dismiss the coaches claims?

http//www.bized.ac.uk/timeweb/crunching/crunch_ex
periment_reviewb.htm
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test of significance

• does our low sample mean __ mean the coach is
  wrong
• OR
• could we have gotten a sample mean __ easily by
  chance

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the null hypothesis

• H0 u 30
• Ha u lt 30
• If the null is true, is our result surprisingly
  low. If yes, thats evidence against the null and
  in favor of the alternative hypothesis.

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hypothesis testing

• Most of you do not know that when Santa was a
  young man he had to take a statistics course.
  When the class started covering hypothesis tests,
  he had a lot of trouble remembering where to put
  the equal sign. He started repeating to himself
  "The equal sign goes in the null hypothesis. The
  equal sign goes in the null hypothesis. The equal
  sign goes in the null hypothesis."
• Eventually Santa had to shorten this phrase to
  make it easier to remember. In fact to this day
  you can still hear him say "Ho, Ho, Ho."

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If the null is true, is our result surprisingly
low.

• sampling distribution of ___

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the p-value

• sampling distr of __ when u is __

p-value
p-value is the prob of getting a result at least
as extreme as the one we got.
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statistically significant

• small p-values are evidence against H0
• RULE OF THUMB a p-value below .05 is called
  statistically significant.
• its unlikely to get a result like ours by chance
  alone.

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one-sided vs. two-sidedalternatives

• Suppose we wanted to test a manufacturers claim
  that there are, on average, 50 matches in a box.
  it would be useful to know if there is likely to
  be____ than 50 matches, on average, in a box.
• Now lets say nothing specific can be said about
  the average number of matches in a box only
  that, if we could reject the null hypothesis in
  our test, we would know that the average number
  of matches in a box is likely to be less than or
  greater than 50.

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One-Sided vs. Two-Sided Tests

• Does the iron concentration of an iron ore exceed
  30 ?
• Is the pizza you ordered from the pizza service
  well done (it could be raw, or burnt, or well
  done)?

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activity

• http//exploringdata.cqu.edu.au/stem_ws1.htm

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P-values and statistical significance
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test statistic

• a stat that estimates the parameter in H0
• H0 u 30
• Ha u lt 30
• ___ estimates ___

p-value
p-value if the null were true
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comment on this p-value
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comment on his p-value
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• Ms. Lisa Monnin is the budget director for the
  New Process Company. She would like to compare
  the daily travel expenses for the sales staff and
  the audit staff. She collected the following
  sample information.
• At the .10 significance level, can she conclude
  that the mean daily expenses are greater for the
  sales staff than the audit staff? What is the
  p-value?
• Sales ()                       131      
  135       146       165       136       142
• Audit ()           130       102       129      
  143       149       120       139
• Having problems finding the p-value unsure of
  the formula.Kathy

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the significance level

• significance level alpha level __
• how much evidence do we need (how low a p-value)
  to reject a claim (reject the null)
• common alpha levels
• __ .05
• __ .01
• if p-value __ alpha level, we say the data are
  statistically significant and we have evidence to
  ___ the null.

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worksheet

• state the parameter
• state the null and alternative hypothesis
• is the alternative one-sided or 2 sided?
• write the test statistic
• graph the distribution when H0 is true
• the mean of sampling distr is ___ sd ___
• write the z test statistic
• find the p-value
• what is the alpha (significance) level?
• is the result statistically significant at alpha
  .05?
• is the result statistically significant at alpha
  .01?
• write your conclusion interpret the p-value

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PROBLEMS

• coffee sales 10.35
• executives blood pressure p573

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tests for a population mean
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form

• estimated value - hypothesized value.
• standard error of the estimate

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how to test

• define your parameter
• state your hypotheses
• calculate the test statistic
• find the p-value
• interpretation and conclusiton.

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one sided alternative

• hypotheses
• H0 u 30
• Ha u 30
• compute the z test statistic
• find the P-value P(Z z)

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two-sided alternative

• hypotheses
• H0 u 30
• Ha u 30
• compute the z test statistic
• find the P-value 2P(Z z)

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what tests do

• tests of significance assess the evidence against
  H0. We never say we have clear evidence that H0
  is true.

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tests with fixed significance levelstests from
CIs/
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how much evidence do I need to reject the null?

• a level of significance alpha tells you how much
• If P lt alpha the outcome of a test is significant
• you can decide whether a result is significant
  without calculating p
• compare your z test statistic to the critical
  value from Table C

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CI and 2-sided tests

• a 2-sided significance test can be carried out
  from a CI with C 1-alpha
• p55510.44

radon
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• Example Students analyze data reported in The
  1992 Statistical Abstract of the United States
  that 30.5 of a sample of 40,000 American
  households own a pet cat (see 19). We ask
  whether this sample provides strong evidence that
  less than one-third of the population of all
  American households owns a cat and then whether
  it provides evidence that much less than
  one-third owns a cat. A significance test answers
  the first question in the affirmative (p-value lt
  .0001), but a confidence interval supplies the
  additional information needed to answer the
  second question in the negative (95 c.i. (.300,
  .310)). Students discern that large sample sizes
  can often lead to statistically significant
  results that are not practically significant.

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using significance tests
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choosing a level of significance

• do this if you must make a decision
• if H0 is an assumption that people have held for
  years, choose (small, large) alpha
• if rejecting H0 will lead to dramatic changes in
  your product, choose (small, large) alpha

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example SAT prep
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about p-value

• use the p-value if you want to describe the
  strength of your evidence
• P-value .0049 vs. P-value .051
• with large samples even tiny deviations from the
  null will be significant
• statistical significance is not the same as
  practical significance.

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antibacterial cream

• p588

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beware multiple analyses

• running 1 test and reaching alpha .05 level is
  ok evidence that you have found something
  significant. Running 20 tests and finding 1
  reaching the alpha .05 level is not.

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beware multiple analyses

• ESP test
• A standard run uses 25 cards. With only 25 cards,
  you must obtain at least 9 hits (correct guesses)
  to give a significant result (plt.05) showing
  possible evidence of ESP.

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teaching methods

• http//www.rossmanchance.com/papers/topten.html

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old faithful

• Example We provide students with data on times
  between eruptions (in minutes) for the Old
  Faithful geyser, originally reported in 2 and
  also presented in 4 and 12. If students
  merely calculate a confidence interval for the
  population mean intereruption time (95 c.i.
  (70.73, 73.90)) without inspecting the data
  first, they fail to notice the pronounced bimodal
  nature of the data with peaks around 55 and 78
  minutes. With this realization students are able
  to describe the inter-eruption times more
  effectively.

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monty hall

• Example In an activity illustrating the famous
  "Monty Hall Problem," we give three playing
  cards, two red and one black, to pairs of
  students. The cards represent prizes behind doors
  used in a game show, one a winner (black) and two
  losers (red). One student (dealer) shuffles the
  cards and holds them facing away from the other
  (contestant). The contestant chooses a card and
  the dealer reveals one of the two remaining cards
  to be red. The contestant is then asked to either
  switch to the remaining card or stay with the
  original choice, in an effort to find the
  (winning) black card. We have the students play
  this game 20 times and perform a test of
  significance to see if the proportion of wins
  using the switching strategy is different from
  1/2. Most students fail to reject this hypothesis
  (power .152). Since 1/2 agrees with many
  students intuition, they do not find this result
  surprising. However, if they continue to play the
  game or reason probabilistically, they discover
  that the actual probability of winning with the
  switch strategy is 2/3. Thus, they learn that the
  original sample size of 20 was not large enough
  to enable them to detect this difference.

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hand span activity
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inference as decision

• date
• hw 10.66, 10.67

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gamehttp//www.stat.psu.edu/resources/InLarge/wl
h_01.htm
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• significance tests H0
• 1. evidence against H0 p-value
• 2. make a decision based on a fixed level, alpha
  if we reject H0 we accept Ha

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Type 1 and Type 2 errors

• Type 1 reject
• Type 2 not reject
• applet

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examples

• Type 1 In other words, it's the rate of false
  alarms or false positives.
• Type 2 In other words, it's the rate of failed
  alarms or false negatives.

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Type 1 and Type 2 errors
H0 is true
Ha is true
dec i s I o n
ACCEPT H0
REJECT H0
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beans

• H0 the sample of beans meets standards
• Ha the sample of beans does not meet standards.
• explain the 2 types of incorrect decisions.
• we can accept a bad bunch
• we can reject a good bunch
• which is more serious?
• which side are you on? consumer? producer?

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example problem
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significance and type 1 error

• the significance level __ P (type I error)
• to find Type II error, we need to know the value
  of a particular alternative
• a high type II error for a particular alternative
  means the test is not sensitive enough to detect
  that alternative

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example, continued
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power of a test

• The power of a test is the probability of
  choosing the alternative hypothesis when the
  alternative hypothesis is correct.
• power 1 P (type II error)

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factors Influencing Power

• Alpha (a). Thus, all other things being equal,
  using an alpha of .05 will result in a more
  powerful test than using an alpha of .01.
• Sample Size (N). The bigger the sample (i.e., the
  more work we do), the more powerful the test.
• Variability. Generally speaking, variability in
  the sample and/or population results in a less
  powerful test.