Review of Top 10 Concepts in Statistics

1
Review of Top 10 Conceptsin Statistics

• NOTE This Power Point file is not an
  introduction, but rather a checklist of topics to
  review

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Top Ten 1

• Descriptive Statistics

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Measures of Central Location

• Mean
• Median
• Mode

4
Mean

• Population mean µ Sx/N (516)/3 12/3 4
• Algebra Sx Nµ 34 12
• Sample mean x-bar Sx/n
• Example the number of hours spent on the
  Internet 4, 8, and 9
• x-bar (489)/3 7 hours
• Do NOT use if the number of observations is small
  or with extreme values
• Ex Do NOT use if 3 houses were sold this week,
  and one was a mansion

5
Median

• Median middle value
• Example 5,1,6
• Step 1 Sort data 1,5,6
• Step 2 Middle value 5
• When there is an even number of observation,
  median is computed by averaging the two
  observations in the middle.
• OK even if there are extreme values
• Home sales 100K,200K,900K, so
• mean 400K, but median 200K

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Mode

• Mode most frequent value
• Ex female, male, female
• Mode female
• Ex 1,1,2,3,5,8
• Mode 1
• It may not be a very good measure, see the
  following example

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Measures of Central Location - Example

• Sample 0, 0, 5, 7, 8, 9, 12, 14, 22, 23
• Sample Mean x-bar Sx/n 100/10 10
• Median (89)/2 8.5
• Mode 0

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Relationship

• Case 1 if probability distribution symmetric
  (ex. bell-shaped, normal distribution),
• Mean Median Mode
• Case 2 if distribution positively skewed to
  right (ex. incomes of employers in large firm a
  large number of relatively low-paid workers and a
  small number of high-paid executives),
• Mode lt Median lt Mean

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Relationship contd

• Case 3 if distribution negatively skewed to left
  (ex. The time taken by students to write exams
  few students hand their exams early and majority
  of students turn in their exam at the end of
  exam),
• Mean lt Median lt Mode

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Dispersion Measures of Variability

• How much spread of data
• How much uncertainty
• Measures
• Range
• Variance
• Standard deviation

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Range

• Range Max-Min gt 0
• But range affected by unusual values
• Ex Santa Monica has a high of 105 degrees and a
  low of 30 once a century, but range would be
  105-30 75

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Standard Deviation (SD)

• Better than range because all data used
• Population SD Square root of variance sigma s
• SD gt 0

13
Empirical Rule

• Applies to mound or bell-shaped curves
• Ex normal distribution
• 68 of data within one SD of mean
• 95 of data within two SD of mean
• 99.7 of data within three SD of mean

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Standard Deviation Square Root of Variance
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Sample Standard Deviation
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Standard Deviation

• Total variation 34
• Sample variance 34/4 8.5
• Sample standard deviation
• square root of 8.5 2.9

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Measures of Variability - Example

• The hourly wages earned by a sample of five
  students are
• 7, 5, 11, 8, and 6
• Range 11 5 6
• Variance
• Standard deviation

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Coefficient of Variation (CV)

• CV Standard Deviation/Mean
• Relative measure of spread
• Example
• Country 1 CV 100/1000 0.10
• Country 2 CV 200/4000 0.05
• Although Country 2 has higher standard
  deviation, Country 1 has higher relative spread

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Graphical Tools

• Line chart trend over time
• Scatter diagram relationship between two
  variables
• Bar chart frequency for each category
• Histogram frequency for each class of measured
  data (graph of frequency distr.)
• Box plot graphical display based on quartiles,
  which divide data into 4 parts

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Top Ten 2

• Hypothesis Testing

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H0 Null Hypothesis

• Population mean?
• Population proportionp
• A statement about the value of a population
  parameter
• Never include sample statistic (such as, x-bar)
  in hypothesis

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HA or H1 Alternative Hypothesis

• ONE TAIL ALTERNATIVE
• Right tail ?gtnumber(smog ck)
• pgtfraction(defectives)
• Left tail ?ltnumber(weight in box of crackers)
• pltfraction(unpopular Presidents
  approval low)

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One-Tailed Tests

• A test is one-tailed when the alternate
  hypothesis, H1 or HA, states a direction, such as

• H1 The mean yearly salaries earned by full-time
  employees is more than 45,000. (?gt45,000)
• H1 The average speed of cars traveling on
  freeway is less than 75 miles per hour. (?lt75)
• H1 Less than 20 percent of the customers pay
  cash for their gasoline purchase. (p lt0.2)

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Two-Tail Alternative

• Population mean not equal to number (too hot or
  too cold)
• Population proportion not equal to fraction (
  alcohol too weak or too strong)

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Two-Tailed Tests

• A test is two-tailed when no direction is
  specified in the alternate hypothesis

• H1 The mean amount of time spent for the
  Internet is not equal to 5 hours. (? ? 5).
• H1 The mean price for a gallon of gasoline is
  not equal to 2.54. (? ? 2.54).

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Reject Null Hypothesis (H0) If

• Absolute value of test statistic gt critical
  value
• Reject H0 if Z Value gt critical Z
• Reject H0 if t Value gt critical t
• Reject H0 if p-value lt significance level (alpha)
• Note that direction of inequality is reversed!
• Reject H0 if very large difference between sample
  statistic and population parameter in H0

Test statistic A value, determined from sample
information, used to determine whether or not to
reject the null hypothesis. Critical value The
dividing point between the region where the null
hypothesis is rejected and the region where it is
not rejected.
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Example Smog Check

• H0 ? 80
• HA ? gt 80
• If test statistic 2.2 and critical value 1.96,
  reject H0, and conclude that the population mean
  is likely gt 80
• If test statistic 1.6 and critical value
  1.96, do not reject H0, and reserve judgment
  about H0

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Type I vs. Type II Error

• Alphaa P(type I error) Significance level
  probability that you reject true null hypothesis
• Beta ß P(type II error) probability you do
  not reject a null hypothesis, given H0 false
• Ex H0 Defendant innocent
• a P(jury convicts innocent person)
• ß P(jury acquits guilty person)

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Type I vs. Type II Error
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Example Smog Check

• H0 ? 80
• HA ? gt 80
• If p-value 0.01 and alpha 0.05, reject H0,
  and conclude that the population mean is likely gt
  80
• If p-value 0.07 and alpha 0.05, do not reject
  H0, and reserve judgment about H0

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Test Statistic

• When testing for the population mean from a large
  sample and the population standard deviation is
  known, the test statistic is given by

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Example

• The processors of Best Mayo indicate on the
  label that the bottle contains 16 ounces of mayo.
  The standard deviation of the process is 0.5
  ounces. A sample of 36 bottles from last hours
  production showed a mean weight of 16.12 ounces
  per bottle. At the .05 significance level, can
  we conclude that the mean amount per bottle is
  greater than 16 ounces?

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Example contd

• 1. State the null and the alternative hypotheses
• H0 ? 16, H1 ? gt 16

2. Select the level of significance. In this
case, we selected the .05 significance level.

• 3. Identify the test statistic. Because we know
  the population standard deviation, the test
  statistic is z.
• 4. State the decision rule.
• Reject H0 if zgt 1.645 ( z0.05)

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Example contd

• 5. Compute the value of the test statistic

• 6. Conclusion Do not reject the null hypothesis.
  We cannot conclude the mean is greater than 16
  ounces.

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Example Using Empirical Rule in Hypothesis
Testing

• H0 ? 80
• HA µ ? 80
• The following information is given
• Computed test statistic (z) -1.14
• Significance level (alpha) 0.05
• Should you reject H0?

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Example contd

• Two tail HA with alpha 0.05 has same critical Z
  value as 95 confidence interval
• Empirical rule, if bell shaped
• 95 of data within two standard deviation of
  mean
• That is, 95 of normal curve between Z -2.0 and
  Z 2.0
• Computed test statistic (Z -1.14) is between
  2.0 and 2.0, so do NOT reject H0

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Top Ten 3

• Confidence Intervals Mean and Proportion

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Confidence Interval

• A confidence interval is a range of values within
  which the population parameter is expected to
  occur.

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Factors for Confidence Interval

• The factors that determine the width of a
  confidence interval are

• The sample size, n
• The variability in the population, usually
  estimated by standard deviation.
• The desired level of confidence.

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Confidence Interval Mean

• Use normal distribution (Z table if)
• population standard deviation (sigma) known and
  either (1) or (2)
• Normal population
• Sample size gt 30

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Confidence Interval Mean

• If normal table, then

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Normal Table

• Tail .5(1 confidence level)
• NOTE! Different statistics texts have different
  normal tables
• This review uses the tail of the bell curve
• Ex 95 confidence tail .5(1-.95) .025
• Z.025 1.96

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Example

• n49, Sx490, s2, 95 confidence
• 9.44 lt ? lt 10.56

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Another Example

• One of SOM professors wants to estimate the mean
  number of hours worked per week by students. A
  sample of 49 students showed a mean of 24 hours.
  It is assumed that the population standard
  deviation is 4 hours. What is the population
  mean?

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Another Example contd

• 95 percent confidence interval for the
  population mean.

The confidence limits range from 22.88 to 25.12.
We estimate with 95 percent confidence that the
average number of hours worked per week by
students lies between these two values.
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Confidence Interval Mean t distribution

• Use if normal population but population standard
  deviation (s) not known
• If you are given the sample standard deviation
  (s), use t table, assuming normal population
• If one population, n-1 degrees of freedom

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Confidence Interval Mean t distribution
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Confidence Interval Proportion

• Use if success or failure
• (ex defective or not-defective,
• satisfactory or unsatisfactory)
• Normal approximation to binomial ok if
• (n)(p) gt 5 and (n)(1-p) gt 5, where
• n sample size
• p population proportion
• NOTE NEVER use the t table if proportion!!

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Confidence Interval Proportion

• Ex 8 defectives out of 100, so p .08 and
• n 100, 95 confidence

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Confidence Interval Proportion

• A sample of 500 people who own their house
  revealed that 175 planned to sell their homes
  within five years. Develop a 98 confidence
  interval for the proportion of people who plan to
  sell their house within five years.

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Interpretation

• If 95 confidence, then 95 of all confidence
  intervals will include the true population
  parameter
• NOTE! Never use the term probability when
  estimating a parameter!! (ex Do NOT say
  Probability that population mean is between 23
  and 32 is .95 because parameter is not a random
  variable. In fact, the population mean is a fixed
  but unknown quantity.)

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Point vs. Interval Estimate

• Point estimate statistic (single number)
• Ex sample mean, sample proportion
• Each sample gives different point estimate
• Interval estimate range of values
• Ex Population mean sample mean error
• Parameter statistic error

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Width of Interval

• Ex sample mean 23, error 3
• Point estimate 23
• Interval estimate 23 3, or (20,26)
• Width of interval 26-20 6
• Wide interval Point estimate unreliable

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Wide Confidence Interval If

• (1) small sample size(n)
• (2) large standard deviation
• (3) high confidence interval (ex 99 confidence
  interval wider than 95 confidence interval)
• If you want narrow interval, you need a large
  sample size or small standard deviation or low
  confidence level.

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Top Ten 4

• Linear Regression

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Linear Regression

• Regression equation
• dependent variablepredicted value
• x independent variable
• b0y-intercept predicted value of y if x0
• b1sloperegression coefficient
• change in y per unit change in x

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Slope vs. Correlation

• Positive slope (b1gt0) positive correlation
  between x and y (y increase if x increase)
• Negative slope (b1lt0) negative correlation (y
  decrease if x increase)
• Zero slope (b10) no correlation(predicted value
  for y is mean of y), no linear relationship
  between x and y

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Simple Linear Regression

• Simple one independent variable, one dependent
  variable
• Linear graph of regression equation is straight
  line

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Example

• y salary (female manager, in thousands of
  dollars)
• x number of children
• n number of observations

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Given Data
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Totals
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Slope (b1) -6.5

• Method of Least Squares formulas not on BUS 302
  exam
• b1 -6.5 given

Interpretation If one female manager has 1 more
child than another, salary is 6,500 lower that
is, salary of female managers is expected to
decrease by -6.5 (in thousand of dollars) per
child
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Intercept (b0)

• b0 44.33 (-6.5)(2.33) 59.5

• If number of children is zero, expected salary is
  59,500

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Regression Equation
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Forecast Salary If 3 Children

• 59.5 6.5(3) 40
• 40,000 expected salary

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Standard Error of Estimate
67
Standard Error of Estimate
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Standard Error of Estimate
Actual salary typically 1,900 away from expected
salary
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Coefficient of Determination

• R2 of total variation in y that can be
  explained by variation in x
• Measure of how close the linear regression line
  fits the points in a scatter diagram
• R2 1 max. possible value perfect linear
  relationship between y and x (straight line)
• R2 0 min. value no linear relationship

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Sources of Variation (V)

• Total V Explained V Unexplained V
• SS Sum of Squares V
• Total SS Regression SS Error SS
• SST SSR SSE
• SSR Explained V, SSE Unexplained

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Coefficient of Determination

• R2 SSR
  SST
• R2 197 .98
  200.5
• Interpretation 98 of total variation in salary
  can be explained by variation in number of
  children

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0 lt R2 lt 1

• 0 No linear relationship since SSR0
  (explained variation 0)
• 1 Perfect relationship since SSR SST
  (unexplained variation SSE 0), but does not
  prove cause and effect

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RCorrelation Coefficient

• Case 1 slope (b1) lt 0
• R lt 0
• R is negative square root of coefficient of
  determination

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Our Example

• Slope b1 -6.5
• R2 .98
• R -.99

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Case 2 Slope gt 0

• R is positive square root of coefficient of
  determination
• Ex R2 .49
• R .70
• R has no interpretation
• R overstates relationship

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Caution

• Nonlinear relationship (parabola, hyperbola, etc)
  can NOT be measured by R2
• In fact, you could get R20 with a nonlinear
  graph on a scatter diagram

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Summary Correlation Coefficient

• Case 1 If b1 gt 0, R is the positive square root
  of the coefficient of determination
• Ex1 y 43x, R2.36 R .60
• Case 2 If b1 lt 0, R is the negative square root
  of the coefficient of determination
• Ex2 y 80-10x, R2.49 R -.70
• NOTE! Ex2 has stronger relationship, as measured
  by coefficient of determination

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Extreme Values

• R1 perfect positive correlation
• R -1 perfect negative correlation
• R0 zero correlation

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MS Excel Output
Correlation Coefficient (-0.9912) Note that you
need to change the sign because the sign of slope
(b1) is negative (-6.5)
Coefficient of Determination
Standard Error of Estimate
Regression Coefficient
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Top Ten 5

• Expected Value

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Expected Value

• Expected Value E(x) SxP(x)
• x1P(x1) x2P(x2)
• Expected value is a weighted average, also a
  long-run average

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Example

• Find the expected age at high school graduation
  if 11 were 17 years old, 80 were 18 years old,
  and 5 were 19 years old
• Step 1 1180596

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Step 2
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Another Example of E(x)

• A news rack has 2 papers left. In past, 20 of
  days you sold both papers, while 50 of days you
  sold one paper. Find expected number of papers
  sold.
• Answer
• First, find P(0) 1 - 0.20 - 0.50 0.30
• E(X) 0(0.30) 1(0.50) 2(0.20) 0.9

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Top Ten 6

• What Distribution to Use?

86
Use Binomial Distribution If

• Random variable (x) is number of successes in n
  trials
• Each trial is success or failure
• Independent trials
• Constant probability of success (p) on each trial
• Sampling with replacement (in practice, people
  may use binomial w/o replacement, but theory is
  with replacement)

87
Success vs. Failure

• The binomial experiment can result in only one of
  two possible outcomes
• Male vs. Female
• Defective vs. Non-defective
• Yes or No
• Pass (8 or more right answers) vs. Fail (fewer
  than 8)
• Buy drink (21 or over) vs. Cannot buy drink

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Binomial Is Discrete

• Integer values
• 0,1,2,n
• Binomial is often skewed, but may be symmetric

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Example of Binomial

• If 60 of all voters in a precinct are Democrats,
  find the probability that a sample of 3 voters
  has (a) all Democrats (b) no Democrats
• Answer for (a)
• P(3) (0.6)(0.6)(0.6)
• Answer for (b)
• P(0) (0.4)(0.4)(0.4)

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Normal Distribution

• Continuous, bell-shaped, symmetric
• Meanmedianmode
• Measurement (dollars, inches, years)
• Cumulative probability under normal curve use Z
  table if you know population mean and population
  standard deviation
• Sample mean use Z table if you know population
  standard deviation and either normal population
  or n gt 30

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t Distribution

• Continuous, mound-shaped, symmetric
• Applications similar to normal
• More spread out than normal
• Use t if normal population but population
  standard deviation not known
• Degrees of freedom df n-1 if estimating the
  mean of one population
• t approaches z as df increases

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Normal or t Distribution?

• Use t table if normal population but population
  standard deviation (s) is not known
• If you are given the sample standard deviation
  (s), use t table, assuming normal population

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Top Ten 7

• P-value

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P-value

• P-value probability of getting a sample
  statistic as extreme (or more extreme) than the
  sample statistic you got from your sample, given
  that the null hypothesis is true

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P-value Example one tail test

• H0 ? 40
• HA ? gt 40
• Sample mean 43
• P-value P(sample mean gt 43, given H0 true)
• Meaning probability of observing a sample mean
  as large as 43 when the population mean is 40
• How to use it Reject H0 if p-value lt a
  (significance level)

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Two Cases

• Suppose a .05
• Case 1 suppose p-value .02, then reject H0
  (unlikely H0 is true you believe population mean
  gt 40)
• Case 2 suppose p-value .08, then do not reject
  H0 (H0 may be true you have reason to believe
  that the population mean may be 40)

97
P-value Example two tail test

• H0 ? 70
• HA ? ? 70
• Sample mean 72
• If two-tails, then P-value
• 2 ? P(sample mean gt 72)2(.04).08
• If a .05, p-value gt a, so do not reject H0

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Top Ten 8

• Variation Creates Uncertainty

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No Variation

• Certainty, exact prediction
• Standard deviation 0
• Variance 0
• All data exactly same
• Example all workers in minimum wage job

100
High Variation

• Uncertainty, unpredictable
• High standard deviation
• Ex 1 Workers in downtown L.A. have variation
  between CEOs and garment workers
• Ex 2 New York temperatures in spring range from
  below freezing to very hot

101
Comparing Standard Deviations

• Temperature Example
• Beach city small standard deviation (single
  temperature reading close to mean)
• High Desert city High standard deviation (hot
  days, cool nights in spring)

102
Standard Error of the Mean

• Standard deviation of sample mean
• standard deviation/square root of n
• Ex standard deviation 10, n 4, so standard
  error of the mean 10/2 5
• Note that 5lt10, so standard error lt standard
  deviation.
• As n increases, standard error decreases.

103
Sampling Distribution

• Expected value of sample mean population mean,
  but an individual sample mean could be smaller or
  larger than the population mean
• Population mean is a constant parameter, but
  sample mean is a random variable
• Sampling distribution is distribution of sample
  means

104
Example

• Mean age of all students in the building is
  population mean
• Each classroom has a sample mean
• Distribution of sample means from all classrooms
  is sampling distribution

105
Sampling

• Sampling Distribution concepts assume
  probability sample
• Probability sample requires calculation of
  probability of being in the sample
• Probability sample more accurate than judgment or
  convenience sample

106
Central Limit Theorem (CLT)

• If population standard deviation is known,
  sampling distribution of sample means is normal
  if n gt 30
• CLT applies even if original population is skewed

107
Top Ten 9

• Population vs. Sample

108
Population

• Collection of all items (all light bulbs made at
  factory)
• Parameter measure of population
• (1) population mean (average number of hours in
  life of all bulbs)
• (2) population proportion ( of all bulbs that
  are defective)

109
Sample

• Part of population (bulbs tested by inspector)
• Statistic measure of sample estimate of
  parameter
• (1) sample mean (average number of hours in life
  of bulbs tested by inspector)
• (2) sample proportion ( of bulbs in sample that
  are defective)

110
Top Ten 10

• Qualitative vs. Quantitative

111
Levels of Measurement

• I. QUALITATIVE
• Nominal No order (Ex color)
• Ordinal Order important (Ex good, fair, poor)
• II. QUANTITATIVE
• Interval Order important (Ex temp, shoe size)
• Ratio Order important AND ratio meaningful (Ex
  20/hr twice as good as 10/hr)

112
Qualitative

• Categorical data
• success vs. failure
• ethnicity
• marital status
• color
• zip code
• 4 star hotel in tour guide

113
Qualitative

• If you need an average, do not calculate the
  mean
• However, you can compute the mode (average
  person is married, buys a blue car made in
  America)

114
Quantitative

• Two cases
• Case 1 discrete
• Case 2 continuous

115
Discrete

• (1) integer values (0,1,2,)
• (2) example binomial
• (3) finite number of possible values
• (4) counting
• (5) number of brothers
• (6) number of cars arriving at gas station

116
Continuous

• Real numbers, such as decimal values (22.22)
• Examples Z, t
• Infinite number of possible values
• Measurement
• Miles per gallon, distance, duration of time

117
Graphical Tools

• Pie chart or bar chart qualitative
• Joint frequency table qualitative (relate
  marital status vs. zip code)
• Scatter diagram quantitative (distance from CSUN
  vs. duration of time to reach CSUN)

118
Hypothesis TestingConfidence Intervals

• Quantitative Mean
• Qualitative Proportion