Last Time Proportions Continuous Random Variables Probabilities Mean and Standard Deviation Male-Female Example Continuous Random Variables Uniform Distribution Normal Distribution

Reading In Textbook Approximate Reading for Today’s Material: Pages 58-64, Approximate Reading for Next Class: Pages 66-70, 61-62, 59-61, 322-326

Proportions Summary: Simple pattern But be careful to keep these straight! Counts, X Proportions Expected Value np p Variance np(1-p) p(1-p)/n Standard Deviation (np(1-p))1/2 (p(1-p)/n)1/2

Big Picture Now exploit constant shape property of Binom’l Centerpoint feels p Spread feels n Big Question: What is common shape?

Normal Distribution Continuous f(x),

Normal Distribution Continuous f(x), that models common shape seen above

Normal Distribution Continuous f(x), that models common shape seen above Goal: Shaped like a mound

Normal Distribution Continuous f(x), that models common shape seen above Goal: Shaped like a mound E.g. sand dumped from a truck

Normal Distribution Continuous f(x), that models common shape seen above Goal: Shaped like a mound E.g. sand dumped from a truck Older (worse) description: bell shaped

Normal Distribution Continuous f(x), that models common shape seen above Like Binomial is indexed family of dist’ns

Normal Distribution Continuous f(x), that models common shape seen above Like Binomial is indexed family of dist’ns Indexed by parameters

Normal Distribution Continuous f(x), that models common shape seen above Like Binomial is indexed family of dist’ns Indexed by parameters μ = mean (average, i.e. center) Greek “mu”

Normal Distribution Continuous f(x), that models common shape seen above Like Binomial is indexed family of dist’ns Indexed by parameters μ = mean (average, i.e. center) σ = standard deviation (spread)

Normal Distribution Nice insight into effect of μ and σ: Applet by Webster West: http://www.stat.tamu.edu/~west/applets/normaldemo1.html

Normal Distribution Nice insight into effect of μ and σ: Applet by Webster West: http://www.stat.tamu.edu/~west/applets/normaldemo1.html

Normal Distribution Nice insight into effect of μ and σ: Applet by Webster West: http://www.stat.tamu.edu/~west/applets/normaldemo1.html

Normal Distribution The “normal density curve” is: usual “function” of

Normal Distribution The “normal density curve” is: circle constant = 3.14…

Normal Distribution The “normal density curve” is: natural number = 2.7…

Normal Curve Mathematics Main Ideas: Basic shape is:

Normal Curve Mathematics Main Ideas: Basic shape is: “Shifted to mu”:

Normal Curve Mathematics Main Ideas: Basic shape is: “Shifted to mu”: “Scaled by sigma”:

Normal Curve Mathematics Main Ideas: Basic shape is: “Shifted to mu”: “Scaled by sigma”: Make Total Area = 1: divide by

Normal Curve Mathematics Main Ideas: Basic shape is: “Shifted to mu”: “Scaled by sigma”: Make Total Area = 1: divide by as , but never

Normal Distribution The “normal density curve” is:

Normal Distribution The “normal density curve” is: Application: fit to data

Normal Distribution The “normal density curve” is: Application: fit to data i.e. Choose μ and σ to fit to histogram

Normal Density Fitting Idea: Choose μ and σ to fit curve to histogram of data,

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data,

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Approach: IF the distribution is “mound shaped”

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Approach: IF the distribution is “mound shaped” & outliers are negligible

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Approach: IF the distribution is “mound shaped” & outliers are negligible THEN a “good” choice of normal model is:

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Example: Melbourne Average Temperature Data

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Example: Melbourne Average Temperature Data Major City in Australia

Normal Density Fitting Idea: Choose μ and σ to normal density to histogram of data, Example: Melbourne Average Temperature Data Major City in Australia

Normal Density Fitting Idea: Choose μ and σ to normal density to histogram of data, Example: Melbourne Average Temperature Data Major City in Australia

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Example: Melbourne Average Temperature Data Major City in Australia, on Arctic Ocean

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Example: Melbourne Average Temperature Data Major City in Australia, on Arctic Ocean Study winter temperatures

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Example: Melbourne Average Temperature Data Major City in Australia, on Arctic Ocean Study winter temperatures Averaged over 1981-1990

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Example: Melbourne Average Temperature Data Analyzed in: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg8.xls

Normal Density Fitting Melbourne Average Temperature Data Data in degrees Centigrade (world standard)

Normal Density Fitting Melbourne Average Temperature Data Data in degrees Centigrade (world standard) Convert to Fahrenheit (interpretable for us)

Normal Density Fitting Melbourne Average Temperature Data Data in degrees Centigrade (world standard) Convert to Fahrenheit (interpretable for us)

Normal Density Fitting Melbourne Average Temperature Data Restrict Attention to Winter Only

Normal Density Fitting Melbourne Average Temperature Data Restrict Attention to Winter Only Since this looks normal (recall mound shaped, no outliers)

Normal Density Fitting Melbourne Average Temperature Data Restrict Attention to Winter Only Since this looks normal (recall mound shaped, no outliers) Will study summer later (needs more powerful methods)

Normal Density Fitting Melbourne Average Temperature Data Histogram (for winter only):

Normal Density Fitting Melbourne Average Temperature Data Histogram (for winter only): Mound shaped

Normal Density Fitting Melbourne Average Temperature Data Histogram (for winter only): Mound shaped No outliers

Normal Density Fitting Melbourne Average Temperature Data Histogram (for winter only): Mound shaped No outliers So fit Normal density

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density Use from data: - Mean

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density Use from data: - Mean - S. D.

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density - Create Grid

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density - Create Grid (usual type 1st two. highlight & drag corner)

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density - Create Grid - Can compute directly from density formula

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density - Create Grid - Can compute directly from density formula Using x, μ, σ, etc.

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density - Create Grid - Can compute directly from density formula - But faster to use NORMDIST

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density - Create Grid - Can compute directly from density formula - But faster to use NORMDIST (parallels BINOMDIST)

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Inputs: - Xgrid

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Inputs: - Xgrid - Data mean

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Inputs: - Xgrid - Data mean - Data S. D.

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Inputs: - Xgrid - Data mean - Data S. D. - Not Cumulative

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Use $ signs for correct drag & drop

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Use $ signs for correct drag & drop - this changes

Normal Density Fitting Melbourne Average Temperature Data Use NORMDIST Use $ signs for correct drag & drop - this changes - these are stable

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density Plot using: - Insert

Normal Density Fitting Melbourne Average Temperature Data Fit Normal density Plot using: - Insert - Line

Normal Density Fitting Melbourne Average Temperature Data

Normal Density Fitting Melbourne Average Temperature Data Mounds shape is good visual fit to data

Normal Density Fitting Melbourne Average Temperature Data Mounds shape is good visual fit to data, thus represents the population effectively

Normal Density Fitting HW: 1.112

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Mound shaped

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Mound shaped Or is it?

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Mound shaped Or is it? Maybe 2 bumps?

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Mound shaped Or is it? Maybe 2 bumps? Or maybe 3?

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Mound shaped Or is it? Maybe 2 bumps? Or maybe 3? Or explainable as natural sampling variation?

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Explainable as natural sampling variation?

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Explainable as natural sampling variation? Useful tool: Hypothesis Test

Melbourne Average Temperature Data Research Corner Melbourne Average Temperature Data Explainable as natural sampling variation? Useful tool: Hypothesis Test (will develop later, need more background)

Normal Distribution The “normal density curve” is:

Normal Distribution The “normal density curve” is: Main Application: Probabilities computed as areas under curve

(continuous probability density) Normal Distribution The “normal density curve” is: Main Application: Probabilities computed as areas under curve (continuous probability density)

Computation of Normal Areas E.g. for

Computation of Normal Areas E.g. for (center)

Computation of Normal Areas E.g. for (spread)

Computation of Normal Areas E.g. for (spread) (# spaces: mean to inflection points)

Computation of Normal Areas E.g. for Area below 1.3 (point on # line)

Computation of Normal Areas E.g. for Area below 1.3 = ?

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute?

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Calculus?

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Calculus?

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Calculus?

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Calculus? Hurdle: no elementary anti-derivative

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Calculus? Hurdle: no elementary anti-derivative i.e. approaches of subst., parts, … all fail

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute?

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Alternate Approach: Numerical Methods

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Alternate Approach: Numerical Methods (e.g. Riemann summation)

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Alternate Approach: Numerical Methods Studied in course on Numerical Analysis

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Convenient Shortcuts

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Convenient Shortcuts (summarized answers):

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Convenient Shortcuts (summarized answers): Historical: table (see Table A in text)

Computation of Normal Areas E.g. for Area below 1.3 = ? How to compute? Convenient Shortcuts (summarized answers): Historical: table (see Table A in text) Excel: function NORMDIST

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1 s.d. = 0.5

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1 s.d. = 0.5

Computation of Normal Areas EXCEL function NORMDIST: Many similarities to BINOMDIST

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1 s.d. = 0.5

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1 s.d. = 0.5

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1 s.d. = 0.5

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 Mean = 1 s.d. = 0.5 Set “Cumulative” to true

Computation of Normal Areas EXCEL function NORMDIST: works in terms of “lower areas” E.g. for Area ≤ 1.3 = = 0.726

Computation of Normal Probs EXCEL Computation: probs given by “lower areas” E.g. for X ~ N(1,0.5) P{X ≤ 1.3} = 0.726

Computation of Normal Probs HW: 1.124 1.121a Notes: “Standard Normal” has mean 0, and s.d. 1 For “shade area”, use Excel to draw density, and then shade by hand

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Computation of Normal Probs EXCEL Computation: probs given by “lower areas” E.g. for X ~ N(1,0.5) P{X ≤ 1.3} = 0.726

Computation of Normal Probs EXCEL Computation: probs given by “lower areas” E.g. for X ~ N(1,0.5) P{X ≤ 1.3} = 0.726 What about other probabilities?

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities?

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? General Rules: Express as “lower probs”

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? General Rules: Express as “lower probs” And use Big Rules of Probability (Same spirit as BINOMDIST before)

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? Nice Visualization: StatsPortal http://courses.bfwpub.com/ips6e.php Resources Statistical Applets Normal Curve

Computation of Normal Probs StatsPortal View E.g. for Area < 1.3 is 0.726

Computation of Normal Probs Computation of upper areas: (use “1 –”, i.e. “not” formula) = 1 -

Computation of Normal Probs Computation of areas over intervals: (use subtraction) = -

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? Class Example 9: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg9.xls

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} P{X ≥ 1.3} P{-4 < X < 1.3} P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} = P{X ≤ 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} = P{X ≤ 1.3} (since P{X = 1.3} = 0)

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} = P{X ≤ 1.3} (since P{X = 1.3} = 0) (since X is a continuous random variable)

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} = P{X ≤ 1.3} = 0.726

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} = P{X ≤ 1.3} = 0.726

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 1.3} P{X ≥ 1.3} P{-4 < X < 1.3} P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3} P{-4 < X < 1.3} P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)} = 1 – P{X < 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)} = 1 – P{X < 1.3} = 1 – 0.726 = 0.274

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3} = 1 – P{not(X ≥ 1.3)} = 1 – P{X < 1.3} = 1 – 0.726 = 0.274

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X ≥ 1.3} P{-4 < X < 1.3} P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} = 0.726

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} = 0.726

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} = 0.726 Note: same answer as above

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} = 0.726 Note: same answer as above Why?

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} = P{X < 1.3} – P{X < -4} = 0.726 Note: same answer as above Why? P{X < -4} ≈ 0 (to 4 decimal places)

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{-4 < X < 1.3} P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = = P{X < 0.7} + P{X > 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = = P{X < 0.7} + P{X > 1.3} = P{X < 0.7} + 1 - P{X > 1.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = = P{X < 0.7} + P{X > 1.3} = P{X < 0.7} + 1 - P{X > 1.3} = 0.549

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = = P{X < 0.7} + P{X > 1.3} = P{X < 0.7} + 1 - P{X > 1.3} = 0.549

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 Alternate approach: use symmetry

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 Alternate approach: use symmetry

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 = 2 * P{X < 0.7}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 = 2 * P{X < 0.7} = 0.549

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 = 2 * P{X < 0.7} = 0.549

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} Visualize on number line:

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} Visualize on number line: 0 1 2

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} Interval centered at 1 0 1 2

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} # (spaces from 1) = 0.3 0 1 2

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} # (spaces from 1) = 0.3 0 1 2 Thus same answer as above

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{X < 0.7 or X > 1.3} = 0.549 P{|X – 1| > 0.3} P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8} Use symmetry here?

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8} Use symmetry here? Careful, asymmetric normal dist

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8} = P{X < -0.8 or X > 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8} = P{X < -0.8 or X > 0.8} = P{X < -0.8} + 1 – P{X < 0.8}

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8} = P{X < -0.8 or X > 0.8} = P{X < -0.8} + 1 – P{X < 0.8} = 0.656

Computation of Normal Probs EXCEL Computation: E.g. for X ~ N(1,0.5), P{X ≤ 1.3} = 0.726 What about other probabilities? P{|X| > 0.8} = P{X < -0.8 or X > 0.8} = P{X < -0.8} + 1 – P{X < 0.8} = 0.656

Computation of Normal Probs HW: 1.125, 1.121 b, c, d 1.136 (0.079, 0.271) 1.137