Chapter 10/ Chapter 11

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Chapter 10/ Chapter 11
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Main Tools

• Pie charts
• Line graphs
• Histograms
• Stem plots
• Box plots
• 5 number summaries

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Representing the Data

• Most everyone is familiar with using graphs to
  display information about data
• Review Handout for more details
• When evaluating a graph look for
• Trends, patterns, deviations, variation
• Read making good graphs in Ch 10
• To describe distributions look at the shape, look
  for center and spread, patterns

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Evaluating Distributions (cont)

• Look for Skewness
• Right skewed
• Left skewed
• Look for symmetry (two halves of the graph are
  mirror images of each other)

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Examples of Bad Graphs

• Evaluate the following graphs and identify the
  problems

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Interpretation of the misleading pictogram
Exports ? Exports3 ?
USA 582 -- 197,137,368 --
Germany 502 16 126,506,008 56
Japan 421 19 (38) 74,618,461 70 (164)
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Stem and Leaf Ex.

• Make a stem and leaf plot from the following data
• 100 110 114 115 115 112 116 118 120
• 33 48 24 43 33 39 22 29 38 20 25 37

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Recap

• A good way to display data is using a graph
• When constructing a graph use good techniques
• When evaluating a graph/distribution look for
  patterns, outliers, variation, skewness etc.
• Try to describe distributions by the shape

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Chapter 12

• Describing Distributions with Numbers
• Statistics-Concepts and Controversies, 6th
  edition, David S. Moore

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Comparing Data

• Median and Quartiles
• One way to describe the spread of the data is to
  give the median and the quartiles
• the median is the midpoint of a distribution,
  half of the
• numbers are larger and half are smaller
• To find the median
• Order the observations from smallest to largest
• If the number of observations n is odd, then the
  median is the middle value. You can find the
  median by counting (n1)/2 observations from the
  top or bottom of the list.
• If the number of observations is even then the
  median is the average of the two middle values.

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Example

• Find the median of the following data sets
• 1 5 9 6 3 7 8 5 1
• 95 68 35 42 96 55 25 35 13 79 28 85

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• Quartiles
• The quartiles are the numbers that divide the
  data into quarters.

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Comparing Data

• Calculating the Quartiles
• Arrange the observations from smallest to largest
  and find the median
• Q1 the median of all observations below the
  median Q2
• Q3 the median of all observations above the
  median Q2

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Examples

• Find the median and quartiles of these data sets
• 1 2 4 7 9 4 6 5 5 2 8 9 0 1 7 8 8
  3 2 6
• 29 84 57 89 46 35 76 83 26 35 15 36
  47 98 63 62 88 87 35 62

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The 5 Number Summary

• Minimum
• Q1
• Median(Q2)
• Q3
• Maximum
• These numbers give a good identification of
  center and spread of the data
• What is the 5 number summary for the two data
  sets we just examined?

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Visuals of the 5 Number Summary

• Box plot
• A box plot contains all 5 numbers in the 5 number
  summary
• To illustrate draw a box plot of past data set

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Other Measures

• Mean another way to measure center of
  distribution also called average (median is
  more robust than the mean)
• sample mean x sum of all observations
• number of observations
• Note population mean

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Other Measures

• Sample Standard Deviation s
• average distance of an observation from the
  mean
• How to find standard deviation
• Find the distance of each observation from the
  mean and then square it
• Add all the squared distances together then take
  the average of these distances using (n-1)
  instead of n (this is the variance)
• Take the square root
• Note population standard deviation
• Best illustrated with examples

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Examples

• Given this data set find the mean and standard
  deviation
• 8 9 6 14 8 3

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More about the Standard Deviation

• s measures the spread about the mean .
• We use s to describe the spread of a distribution
  only when we use the mean to describe the
  center.
• s 0 only when there is no spread.
• This happens only when all the observations have
  the same value. Otherwise, s gt 0.
• As the observations become more spread out about
  their mean, s gets larger.

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Choosing Numerical Descriptions

• The five-number summary is the best short
  description for most distributions.
• The mean standard deviation are harder to
  understand but are more common.
• We must keep in mind that the mean is greatly
  influenced by a few extreme observations the
  median is not.

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• Symmetric Distributions The mean and the median
  are about the same.
• Skewed Distributions The mean runs away from
  the median toward the long tail.
• ie Skewed Right Median lt Mean
• Standard Deviation It is pulled up by outliers
  or long tails. The quartiles are much less
  sensitive to a few extreme observations.

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Chapter 13

• Normal Distributions
• Statistics-Concepts and Controversies, 6th
  Edition, David S. Moore

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Strategies for Exploring Data

• Plot the data
• Histogram/(stem and leaf plot)
• Best to visualize shape
• Look for the overall pattern for striking
  deviations such as outliers
• Choose either the five-number summary or the mean
  standard deviation to briefly describe the
  center and spread in the data.
• Sometimes, the overall pattern of a large number
  of observations is so regular that we can
  describe it by a smooth curve.

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• Histograms of Large Data Sets (Density Curves)
• Remark A relative frequency histogram for an
  infinitely large data set looks like a smooth
  curve.
• Notes
• 1.) By Convention, area (not height)
• measures relative frequency.
• 2.) Area under the entire curve 1

relative frequency
x
a
b
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Center and Spread

• Areas under a density curve represent proportions
  of the total of observations.
• Median center point such that each half has
  equal areas
• Quartiles divide area under the curve into
  quarters
• Mean balance point (pt at which the curve would
  balance if made of solid material)
• Symmetric Density Curve mean and median are the
  same point.
• Curve that is Skewed to the Right the mean is
  larger than the median.
• Recall that the mean is affected by outliers more
  than the median. Hence, the few high
  observations pull the mean towards the tail.
• Curve that is Skewed to the Left the mean is
  smaller than the median.

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The Normal Distribution

• Symmetric, bell-shaped curves with the following
  properties
• A specific normal curve is completely described
  by giving its mean and standard deviation.
• The mean determines the center.
• it is symmetric about the mean
• The standard deviation determines the shape of
  the curve.
• It is the distance from the mean to the
  change-of-curvature points on either side.

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The Empirical Rule

• A.K.A. The 68 95 99.7 Rule
• In any normal distribution,
• approximately 68 of the observations fall within
  one standard deviation of the mean.
• approximately 95 of the observations fall within
  two standard deviations of the mean.
• approximately 99.7 of the observations fall
  within three standard deviations of the mean.

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Visualization of the Empirical Rule
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Standard Scores Z

• One way of describing the location of any
  observation in a normal distribution is to
  calculate its standard score. This score
  indicates how many standard deviations an
  observation lies above or below the mean.

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• Example 1
• The mean of a data set is 50 and the standard
  deviation is 12. What is the standard score if I
  have an observation of 25?
• Example 2

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Percentiles

• The cth percentile of a distribution is a value
  such that c percent of the observations lie below
  it and the rest lie above.
• Think of the quartiles
• We can use Appendix B to find the percentiles of
  standard scores

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Example

• Using table B and the previous Example 2, what is
  the percentile for the standard score for our
  observation of 37?
• What is the percentile of 24?

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Examples

• Suppose that on a statistics test the mean grade
  is 75 and the standard deviation is 8. Assume
  that the scores on the test vary according to a
  distribution that is approximately normal.
• Sixty-eight percent of the data fall into what
  range?
• Almost all (99.7) of the scores fall in what
  range?
• How high did the top 2.5 of the class score?

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Example

• A set of SAT scores has a mean of 890 and a
  standard deviation of 120. Assume the data are
  bell-shaped.
• Lanes SAT score is 1130. Calculate her standard
  score.
• Approximately what proportion of students
  received a score higher than Lanes?
• Mikes SAT score is 770. Calculate his standard
  score.
• Approximately what proportion of students
  received a score lower than Mikes?
• Kevins SAT score is 1010. Calculate his
  standard score.
• Approximately what proportion of students
  received a score lower than Kevins?
• Approximately what proportion of students
  received a score between 650 and 890?

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Example

• The length of human pregnancies from conception
  to birth varies according to a distribution that
  is approximately normal with mean 266 days and
  standard deviation 16 days. Use this information
  to answer the questions below.
• Between what values do the lengths of the middle
  99.7 of all pregnancies fall?
• About what percent of the pregnancies are less
  than 234 days?
• How long are the longest 16 of all human
  pregnancies?
• What percent of these pregnancies last less than
  258 days?
• What percent of these pregnancies last more than
  290 days?
• What percent of these pregnancies last between
  258 and 290 days?
• How long is a pregnancy which falls into the
  13.57 percentile?

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Example

• The SAT math test among Kentucky high school
  seniors in a recent year were normally
  distributed with mean 440 and standard deviation
  60. Use this information to answer the following
  questions
• Into what percentile would a student with a score
  of 398 have fallen?
• What score would a student have achieved if she
  fell into the top 3.6?

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Example

• Suppose that the average height for adult males
  is normally distributed with a mean of 70 inches
  and a standard deviation of 2.5 inches.
• What percentile does a man who is 68 inches fall
  into?
• What percentile does a man who is 73 inches fall
  into?
• What proportion of men are shorter than 74
  inches?
• What percent of men are taller than 72 inches?
• How tall is a man in the 9.68 percentile?
• How tall is a man who has 8 of all men taller
  than him?
• Determine the percentage of men falling between
  69.25 inches and 73.5 inches.

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Example

• In the summer, a grocery store brings in a large
  supply of watermelons. The mean weight in pounds
  is 22. The variance is 16.
• What percent of watermelons weigh between 18 and
  20 pounds?
• What percent of watermelons weigh less than 18
  pounds?
• What percent of watermelons weigh more than 17
  pounds?
• What percent of watermelons weigh more than 30
  pounds?

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Chapter 14

• Describing Relationships
• Scatter plots and Correlation
• Statistics-Concepts and Controversies, 6th
  Edition, David S. Moore

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Scatterplot

• Show the relationship between two quantitative
  variables measured on the same individual. Each
  individual appears as a point in the plot, fixed
  by the values of both variables for that
  individual.
• When applicable, put the explanatory variable on
  the horizontal axis and the response variable on
  the vertical axis.

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Interpretation

• Look for an overall pattern and for striking
  deviations from that pattern.
• Describe the pattern by form, direction,
  strength of the relationship.
• Form Look for clusters and for the shape
  (i.e. curved/linear/nothing/other)
• Direction Is it positively associated, negative
  associated, or neither?
• Strength Determined by how closely the points
  follow a clear form.
• Locate outliers

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Linear Relationships

• Why?
• Correlation Describes the direction strength
  of a straight-line relationship between 2
  quantitative variables. The notation is r.

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Understanding Correlation

• Some important facts about Correlation
• Positive r indicates positive association.
• Negative r indicates negative association.
• Always between -1 and 1.
• The closer r is to1 or -1, the stronger the
  relationship.
• Does not change if we change units
• Ignores distinction between explanatory
  response variable.
• Measures the strength of ONLY straight-line
  associations between 2 variables.
• Strongly affected by a few outlying observations.

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Examples

• Would you expect these correlations to be
  positive, negative or nothing (i.e. r 0)
• The heights and weights of adult men
• The age of secondhand cars and their prices
• The weight of new cars and their gas mileages in
  miles per gallon
• The heights and the IQ scores of adult men
• The heights of husbands and the heights of their
  wives
• The number of work-hours in safety training and
  the number of work-hours lost due to accidents

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Chapter 15

• Describing Relationships Regression, Prediction
  and Causation
• Statistics-Concepts and Controversies, 6th
  Edition, David S. Moore

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Regression

• When we have a scatterplot with a linear
  relationship, we are often interested in
  summarizing the overall pattern. We can do this
  by drawing a line on the graph. This type of
  line is called a regression line. A regression
  line is a straight line that describes how a
  response variable y changes as an explanatory
  variable x changes. We are often interested in
  using this line to predict a value of y for a
  given value of x.

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Regression Lines

• In order to draw a regression line, we must have
  a regression equation.
• Using the data we are given we come up with the
  best regression equation which will result in the
  best regression line. The best regression line
  is the one that comes the closest to the data
  points in the vertical direction. There are many
  ways to make this distance as small as
  possible.
• least-squares method the most common method.
  The least-squares regression line of y on x is
  the line that makes the sum of the squares of the
  vertical distances of the data points from the
  line as small as possible.
• Note We will not actually perform the
  least-squares method. Instead, we are interested
  in being able to use the resulting line.

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Background of a Line

• Recall from past math courses that the equation
  of a line has the form y a bx.
• We write the regression equation in the form y
  a bx.
• y represents the (response) variable on the
  y-axis, or the vertical axis.
• x represents the (explanatory) variable on the
  x-axis, or the horizontal axis.
• b represents the slope. The slope tells us how
  much increase there is in the y for every one
  unit increase in the x. In other words, b 3
  would mean that if the x variable increases one
  unit, then the y variable increases 3 units.
• a represents the y-intercept, or the point where
  the line crosses the vertical axis.
• We can use this equation to predict the value of
  y for any given x.

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Example

• The regression line between the age of a wife and
  the age of a husband is given by y 3.6 .97x
  where x is the wifes age in years and y is the
  husbands age in years.
• If a wife is 30 years old, then we would estimate
  that her husband is approximately 32.7 years old.

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Correlation and Regression

• Recall that correlation measures the strength and
  direction of a linear relationship. We now know
  that regression is what is used to draw the line
  representing this relationship.
• Correlation and regression are closely connected.
• Both correlation and regression are strongly
  affected by outliers.
• The usefulness of the regression line depends on
  the correlation between the two variables.
• We use the square of the correlation, called R -
  squared. It is the fraction of the variation in
  the values of y that is explained by the
  least-squares regression of y on x.
• The idea is that when there is a straight-line
  relationship, some of the variation in y is
  accounted for by the fact that as x changes, it
  pulls y along with it.
• Ex. If r .6, then .36, meaning that
  roughly 36 of the variation is accounted for by
  the straight-line relationship.

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Prediction

• Prediction is based on fitting some model to a
  set of data.
• All of the models we will be looking at involve
  only a linear relationship between one
  explanatory and one response variable. Other
  prediction methods use more elaborate models.
• Prediction works best when the model fits the
  data closely.
• The closer the data actually follows a linear
  pattern, the better the prediction will be.
• Prediction outside the range of the available
  data is risky.
• It is not a good idea to use a regression
  equation to predict values far outside the range
  where the original data fell. In other words,
  the data used to calculate the regression
  equation for the relationship between a husbands
  and a wifes age given above was comprised of men
  and women ranging from about 20 to about 65.
  Therefore, it would not be a good idea to use
  this equation to estimate the age of a womans
  husband if she was 75. We should only use the
  equation for a minor extrapolation beyond the
  range of the original data.
• ie you wouldnt use a young childs growth to
  predict how tall they will be at age 40

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Causation

• Watch Out! A strong relationship between two
  variables does not always mean that changes in
  one variable cause changes in the other.
• The relationship between two variables is often
  influenced by other variables lurking in the
  background.

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Funny Examples

• A strong correlation has been found in a certain
  city in the northeastern United States between
  weekly sales of hot chocolate and weekly sales of
  facial tissues.
• Can we conclude causation ?
• There is a strong correlation between the number
  of women in the work force versus the number of
  Christmas trees sold in the United States for
  each year between 1930 and the present.
• Can we conclude causation ?

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Why Two Variables Could Be Related

• The explanatory is the direct cause of the
  response variable.
• Example Variable A is the pollen count from
  grasses and variable B is the percentage of
  people suffering from allergy symptoms, measured
  over a year. A is the direct cause of B.
• The response variable is causing a change in the
  explanatory variable.
• Example In a study in Resource Manual, it was
  noted that divorced men were twice as likely to
  abuse alcohol as married men. The authors
  concluded that getting divorced caused alcohol
  abuse. But, it is just as reasonable to assume
  that alcohol abuse causes divorce.

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Why Two Variables Could Be Related

• The explanatory variable is a contributing, but
  not the sole, cause of the response variable.
• Example Consider the relationship between hours
  studied per day and grade point average.
  Studying increases grade point average, but it is
  also reasonable that a desire to do well in
  school means that a person studies more and that
  their grade point average is high.
• Confounding variables may exist.
• Example Meditation was found to be related to
  lower levels of an aging factor. It may be that
  meditation does indeed slow aging, but the
  influence of other factors cannot be separated in
  terms of their effect on aging, like a general
  concern for ones well-being.

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Why Two Variables Could Be Related

• Both variables may result from a common cause.
• Look at the example under reason two. Divorce
  and alcohol abuse are related. It may be that
  both result from an unhappy relationship, for
  whatever reason.
• Both variables are changing over time.
• Example The number of divorces and the number
  of suicides have both increased dramatically
  since 1900. This does not mean that divorces are
  causing suicides. All such statistics increase
  as the population increases.

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Why Two Variables Could Be Related

• The association may be nothing more than
  coincidence.
• Example Paulos (1994) relates this story.
  Consider the near panic that ensued last year
  when a guest on a national talk show blamed his
  wifes recent death from brain cancer on her use
  of a cellular telephone. The man alleged that
  there was a causal connection between his wifes
  frequent use of their cellular phone and her
  subsequent brain cancer. It is then noted that
  if brain cancer rates among cellular phone users
  were equal to the rate for the general population
  there would be about 700 cases a year, yet only a
  few have come to light.

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Causation

• The best evidence for causation comes from
  randomized comparative experiments.
• The only legitimate way to try to establish a
  causal connection statistically is through the
  use of designed experiments.
• Evidence of a possible causal connection.
• 1.There is a reasonable explanation of cause and
  effect.
• 2.The connection happens under varying
  conditions.
• 3.Potential confounding variables are ruled out.
• Other things to keep in mind Data from an
  observational study in the absence of any other
  evidence cannot be used to establish causation.

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Example

• For a certain type of automobile, yearly repair
  costs in dollars (Y) are approximately linearly
  related to the age in years (X) of the car. A
  sample of cars which were 1 to 10 years old
  yielded a regression line of
• y 69.7548 9.5221x.
• Estimate the repair costs of a 6-year-old car and
  a 3-year-old car.
• What is the slope of the regression line?
• Interpret what the slope means for this
  regression line.
• Is the correlation between repair cost and age
  positive or negative? Support your answer.

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Chapter 21

• What is a Confidence Interval?
• Statistics-Concepts and Controversies, 6th
  Edition, David C. Moore

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What is a Confidence Interval?

• Statistical Inference Draws conclusions about a
  population on the basis of data from a sample.
• Recall parameters tell us something about
  populations whereas, statistics address samples
• We will use a sample statistic to estimate a
  population parameter.

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Estimating Sample Statistics

• p-hat will be the statistic that we use to
  estimate the true population proportion p, the
  parameter we wish to estimate.

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Examples

• Suppose I survey 200 UK students and ask if they
  have studied in the library in the past week.
  Suppose 150 answer yes.
• What is the sample proportion?
• What is the sample?
• What is the population?
• I take a survey of 500 Gainesville, FL residents,
  who are over 18 and registered to vote, and ask
  if they are planning to vote in the next
  election. Suppose 450 answer yes.
• What is the sample proportion?
• What is the sample?
• What is the population?

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Confidence Intervals

• 95 Confidence Interval an interval calculated
  from sample data that is designed to contain the
  true population parameter in 95 of all samples.
  (Notice the confidence is in the method!)
• Using repeated sampling!

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Estimating Confidence

• We will use p-hat from an SRS to estimate the
  true population proportion p.
• What we want to ask ourselves is What would
  happen if we took many samples?
• Note p-hat varies from sample to sample.
  Sampling variability, however, has a clear
  pattern in the long run, a pattern that is well
  described by a normal curve centered around p.
• Sampling Distribution Distribution of values
  taken by the statistic in all possible samples of
  the same size from the same population.
• We have briefly discussed this before

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Estimating Confidence cont. Paraphrase of CLT
(Important)

• Take an SRS of size n from a large population
  that contains a proportion parameter p of
  successes. Let be the sample proportion of
  successes. If the sample is large enough, then
• The sampling distribution of is
  approximately normal.
• The mean of the sampling distribution is p.
• The standard deviation of the sampling
  distribution is

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Example

• We ask 500 adults if they jog. Of the people we
  asked 120 responded yes. Suppose we know that
  15 of all adults jog.
• What is p-hat?
• What is the sampling distribution of ?
• Find the ranges for the middle 95 of all
  samples.

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Relationship to Confidence Interval

• 95 of all samples give an outcome of such
  that the population truth p is captured by the
  interval from

- 2(s.d.) to
2(s.d.).
This is written as In practice, however, this
is not very helpful. In order to find the exact
standard deviation, we must know p, but if we
knew p, we would not be doing the sampling in the
first place. Since in practice we do not know p,
we will use as our estimate for p.
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Creating A Confidence Interval

• 95 Confidence Interval For a Proportion
• Choose an SRS of size n from a large population
  that contains an unknown proportion p of
  successes. Call the proportion of successes in
  this sample
• An approximate 95 confidence interval for the
  parameter p is

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Creating an Interval Cont.

• We can use other levels of Confidence.
• A level C confidence interval has two parts.
• The Interval calculated from the data.
• The Confidence Level, C, which gives the
  probability that the interval will capture the
  true parameter value in repeated samples.
• (So, a 95 confidence interval means that
  95 of the time, the method produces an interval
  that does capture the true parameter.)
• Z is called the critical value of the normal
  distribution.
• Table 21.1 (pg. 435) gives you Z for a
  particular level of confidence

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General Formula for a Confidence Interval

• .

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Examples

• Suppose we flip a poker chip 100 times. One side
  of the poker chip is black and the other side is
  red.
• Suppose we get 48 reds. Construct a 95
  confidence interval for the overall proportion of
  reds.
• Suppose we flip another poker chip and get only
  35 reds. What is the 95 confidence interval
  now?
• What conclusions can you make using the above
  confidence intervals?
• Suppose we flip a fair coin 100 times. Describe
  the sampling distribution of the proportion of
  heads. Apply the Empirical rule to this
  distribution.

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More Examples!

• Suppose we took two different SRS of 500 adults
  and asked them if they jogged. In the first
  sample, 70 people said yes. In the other sample,
  100 people said yes. Find a 95 confidence
  interval for the true population proportion p
  using each of the sample statistics.

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Example

• UKs student government decides to examine the
  proportion of students who eat at on-campus
  restaurants. Of the 250 students surveyed, 175
  eat on campus.
• Say in words what the population proportion p is
  for this situation.
• Find a 95 confidence interval for the proportion
  of all students who eat on campus.
• Interpret the resulting interval in words that a
  statistically naïve reader would understand.
• Given your interval in part (c), should you
  conclude that the majority of all students eat on
  campus? Why or why not?

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Example

• Construct a 90 and a 99 confidence interval for
  the proportion of all students who eat on campus
  using the sample result in the previous example.
  How do these intervals compare to the 95
  confidence interval?

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Chapter 22

• What is a Test of Significance?
• Statistics-Concepts and Controversies, 6th
  Edition, David C. Moore

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Statistical Tests

• If a friend tells you I can run a 5K race in
  under 20 minutes but youre friends time at
  the 6 5K races you both have run in is over 27
  minutes. What would you be inclined to believe?
  Why?

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Tests of Significance

• Determine the hypotheses
• Null Hypothesis an assumption concerning the
  value of the population parameter being studied
  (usually represents no effect, no change, no
  difference, etc.)
• Notation H0
• Note The null always contains the equality
• Alternative Hypothesis a statement that
  specifies an alternative set of possible values
  for the population parameter that is not included
  in the null hypothesis (states the result for
  which we hope to find evidence)
• Notation HA (or H1)
• Note The null and alternative always contradict
  eachother
• Note The null hypothesis may or may not be true.
  We will carry out a study and then determine if
  we have strong enough evidence to conclude that
  the null hypothesis is false (meaning our
  evidence suggests that HA is true).

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Examples

• A psychology text states that 10 of the
  population is left-handed. You do not know
  whether the proportion of left-handers is more or
  less than .1. What null and alternative
  hypotheses should you test?
• Note the differences in hypotheses between
• Not equal to HA p ? .10
• Greater than HA p gt .10
• Less than HA p lt .10

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Tests of Significance

• Obtain a simple random sample of n observations
  from the desired population and calculate the
  observed sample statistic.
• For example, if we want to test something about a
  population proportion (p), then we would
  calculate the sample proportion .
• If the null hypothesis is true, our sample
  proportion can be approximately described by a
  normal curve with
• We use this mean and standard deviation to
  construct the test statistic (z) for our observed
  sample statistic

87
Evaluating a Test

• Determine the strength of your evidence.
• The evidence is strong if the outcome we observe
  would rarely occur assuming the null hypothesis
  is true (meaning it is more probable that the
  alternative hypothesis is true).
• The evidence is weak if the outcome we observe
  has a high probability of occurring assuming the
  null hypothesis is true.
• We measure the strength of the evidence by
  calculating a P-value.
• p-value the probability of obtaining a sample
  outcome at least as extreme as the actual
  observed outcome, assuming the null is true.
  (Know this defintion!)
• The smaller the p-value, the stronger the
  evidence is against H0.
• (You may also think of the p-value as describing
  the risk of making a mistake if we wrongly reject
  the null.)

88
Evaluating a Test (cont)

• Draw a conclusion.
• If the p-value is small, then we reject H0 in
  favor of HA.
• If the p-value is large, then we fail to reject
  H0, meaning we cannot conclude H0 is false
• You may NEVER conclude that the null is true.
  Unfortunately, you CANNOT be certain that you
  have made the correct conclusion. i.e. we would
  not state we accept the null

89
More about conclusions

• we decide in advance how small the p-value must
  be in order to conclude that we have strong
  evidence against H0.
• The value we choose is called the significance
  level (written as a ).
• If the p-value is as small as or smaller than a,
  then we say that the data is statistically
  significant at level a (meaning that the observed
  outcome would rarely occur by chance).
• a .05 is the most common. When assuming H0 is
  true, this means that the data must give strong
  evidence that this result would occur by chance
  no more than 5 of the time.
• a .01 requires stronger evidence against H0.
• Statistically significant does NOT necessarily
  mean practically important. It only means not
  likely to happen by chance alone.
• Giving the p-value is ALWAYS more informative
  than just stating if the results are
  statistically significant or not.

90
The p-value approach

• Advantages to this Approach
• When the P-value is reported, the decision of
  whether or not to reject the null hypothesis is
  left up to the reader.
• For example, suppose a p-value of .03 is
  reported. If you, the reader, think that a 5
  level of significance (a .05) is sufficient,
  then you would choose to reject the null
  hypothesis in favor of the alternative
  hypothesis. If, however, a second reader thinks
  that a 5 level of significance is insufficient
  and would rather use a .01, then he or she
  would fail to reject the null hypothesis.

91
Publishing Our Results

• p-values are very often reported when describing
  the results of studies in many fields.
  Therefore, it is very important to understand
  what they are telling you.
• Example The financial aid office of a university
  asks a sample of students about their employment
  and earnings. The report says, For academic
  year earnings, a significant difference ( p-value
  .038) was found between the sexes, with men
  earning more on the average.
• Interpretation If there really is no difference
  in academic year earnings between the sexes, then
  we would have seen a difference this big or
  bigger in only 3.8 of all samples. (i.e. There
  is only a 3.8 chance that these results occurred
  by chance alone.)

92
Examples

• An economist states that 10 of a citys labor
  force is unemployed. Suppose you think that this
  estimate is too low. What null and alternative
  hypotheses should you test?
• A state legislature says that it is going to
  decrease its funding of the state university
  because, according to its sources, 36 of the
  graduates move out of the state within 3 years of
  graduation. As a faculty member at the
  university, you want to show that the percent of
  graduates who move out of state is less than 36.
  What null and alternative hypotheses should you
  test?

93
Examples

• Suppose you think that the proportion of people
  who wear contact lenses that experience no
  difficulty is less than 80. You wish to conduct
  a hypothesis test to determine if you are
  correct.
• What null and alternative hypotheses should you
  use?
• Suppose you carry out the above hypothesis test
  on a sample of 200 students. You obtain a sample
  proportion of .745 and get a P-value of .0262.
  Carefully explain what this P-value means in this
  particular situation.

94
Examples

• The is testing a new method to teach soldiers to
  shoot a rifle. A larger proportion of soldiers
  pass the marksmanship test after 3 days of
  training using the new method. (74 vs. 70,
  P-value .0228)
• Set up the appropriate hypothesis.
• What does this p-value lead you to conclude?
• A farmer planted wheat in 100 plots of land. On
  50 plots, he used fertilizer by company 1 and on
  the other 50 plots, he used fertilizer by company
  2. The average yield on the plots is different.
  (P-value .0004, fertilizer by co. 1 yielded 14
  higher)
• What does this p-value lead you to conclude?

95
Examples

• Verify that the P-value in the problem concerning
  people who wear contact lenses is correct.
• Is the result in part (b) of the problem
  statistically significant at the 5 level? At
  the 1 level?
• Going back to the state legislature problem,
  suppose you obtain a random sample of 160
  graduates and find that 40 moved out of state
  within 3 years of graduation. Calculate the
  corresponding P-value.

96
Example

• In a random sample of 200 walnut panels, 32 had
  major flaws. Is this sufficient evidence for
  concluding that the proportion of walnut panels
  that contain major flaws is greater than .1?
• Use a .05.

97
Example

• According to Myers-Briggs estimates, about 82 of
  college student government leaders are
  extroverts. (Source Myers-Briggs Type
  Indicator Atlas of Type Tables.) Suppose that a
  Myers-Briggs personality preference test was
  given to a random sample of 73 student government
  leaders attending a large national leadership
  conference and that 56 were found to be
  extroverts. Does this indicate that the
  population proportion of extroverts among college
  student government leaders is not 82? Use a 5
  level of significance.

98
Example

• The publisher of a magazine is told that the
  percentage of subscribers to the magazine that
  are younger than 36 years is 60. The publisher
  thinks that the percentage is different.
• What null and alternative hypotheses should you
  use?
• Suppose you carry out the above hypothesis test
  on a sample of 150 subscribers. You obtain a
  sample proportion of 0.51.
• Calculate the correct test statistic and p-value
  for this problem.
• What should the publisher conclude?

99
Chapter 23

• Use and Abuse of Statistical Inference
• Statistics-Concepts and Controversies, 6th
  Edition, David C. Moore

100
Using Inference Wisely

• The design of the data production matters.
• For our confidence interval and test for a
  proportion p, we must have a simple random sample
  (SRS).
• If we have poor data collection methods our
  results may be invalid.

101
Know how confidence intervals behave

• The confidence level says how often the method
  catches the true parameter under repeated
  sampling.
• We are confident in the method! i.e. The method
  works 95 of the time for a 95 confidence
  interval.
• We never know if our particular interval actually
  contains p.
• The higher the confidence level, the wider the
  interval.
• Larger samples give shorter intervals. Notice
  the formula

102
The advantages of confidence intervals

• Confidence intervals can be more informative than
  tests because they actually estimate a population
  parameter.
• They are also easier to interpret.
• It is good practice to give confidence intervals
  whenever possible.

103
Hypothesis Tests

• Know what statistical significance says.
• A significance test answers only one question
  How strong is the evidence that the null
  hypothesis is not true?
• The p-value measures how unlikely our data would
  be if the null is true
• We never know whether the hypothesis is true for
  the specific population.

104
Hypothesis Test cont.

• Know what your methods require.
• Our test and confidence interval for a proportion
  p require that the population be much larger than
  the sample.
• They also require that the sample itself be
  reasonably large.
• Keep in mind the effects of sample size on
  hypothesis tests.

105
When does testing for significance make sense?

• Significance tests work when we form a hypothesis
  and then wait for the data.
• Look back and take the best isnt a suitable
  foundation for a significance test.

106
The woes of significance tests.

• Larger samples make tests of significance more
  sensitive whereas tests of significance based on
  small samples are not sensitive.
• When reporting a p-value, you should also include
  the sample size and the statistic that describes
  the sample outcome. Reason The p-value depends
  strongly on the size of the sample and the truth
  about the population.

107
Significance at the 5 level isnt magical

• There is no sharp border between significant
  and insignificant, only increasingly strong
  evidence as the p-value decreases.
• This means that there is no practical distinction
  between a p-value of 0.051 and a p-value of 0.049.

108
Beware of Searching for Significance

• The main goal is to find a significant effect
  that you were looking for.
• This does not mean go out and run a hundred tests
  and then report which ones you found that were
  significant.
• Remember at a 5 significance level out of a 100
  tests on average 5 of them would be found to be
  significant on chance alone.