Ch 7 Continuous Probability Distributions Chapter 7 Continuous Probability Distributions Statistics Are Fun & Statistics Is Fun!

Ch 7 Goals Understand the difference between discrete and continuous distributions Compute the mean and the standard deviation for a uniform distribution Compute probabilities using the uniform distribution Statistics Are Fun & Statistics Is Fun!

Goals List the characteristics of the: Define and calculate z values Normal probability distribution Standard normal probability distribution Define and calculate z values Use the standard normal probability distribution to find area: Above the mean Below the mean Between two values Above one value Below one value Use the normal distribution to approximate the binomial probability distribution Statistics Are Fun & Statistics Is Fun!

Probability Distribution Ch 7 Probability Distribution A listing of all the outcomes of an experiment and the probability associated with each outcome Probability distributions are useful for making probability statements concerning the values of a random variable Our goal is to find probability between two values: Example: What is the probability that the daily water usage will lie between 15 and 25 gallons? A: 68% Statistics Are Fun & Statistics Is Fun!

Probability Distribution Ch 7 Probability Distribution Discrete Probability Distributions (Chapter 6) Based On Discrete Random Variables We looked at: Binomial Probability Distribution Continuous Probability Distribution (Chapter 7) Based On Continuous Random Variables: We will look at: Uniform Probability Distribution Normal Probability Distribution Statistics Are Fun & Statistics Is Fun!

Continuous Probability Distributions Ch 7 Continuous Probability Distributions These continuous probability distributions will be all about  Area!!!! Statistics Are Fun & Statistics Is Fun!

Continuous Probability Distribution: Ch 7 Continuous Probability Distribution: Uniform Probability Distribution Within the interval 15 to 25 minutes, the time it takes to fill out a typical 1040EZ tax return at a VITA site tends to follow a uniform distribution Random Variable is time (only possibilities within the interval) Each value has same probability Normal Probability Distribution The weight distribution of a manufactured box of cereal tends to follow a normal distribution Random Variable is box weight and will cover all possibilities Each value has different probability Statistics Are Fun & Statistics Is Fun!

Uniform Probability Distribution Ch 7 Uniform Probability Distribution Distributions shape is rectangular Minimum value = a Maximum value = b a and b imply a range Height of the distribution is constant (uniform) for all values between a and b Implies all values in range are equally likely Statistics Are Fun & Statistics Is Fun!

Ch 7 Statistics Are Fun & Statistics Is Fun!

What is the standard deviation of the wait time? Ch 7 What is the mean wait time? Suppose the time that you wait on the telephone for a live representative of your phone company to discuss your problem with you is uniformly distributed between 5 and 25 minutes. a + b 2 m = 5+25 2 = = 15 What is the standard deviation of the wait time? (b-a)2 12 s = (25-5)2 12 = = 5.77 Statistics Are Fun & Statistics Is Fun!

The area from 10 to 25 minutes is 15 minutes. Thus: Ch 7 What is the probability of waiting more than ten minutes? The area from 10 to 25 minutes is 15 minutes. Thus: P(10 < wait time < 25) = height*base = 1 (25-5) *15 = .75 Statistics Are Fun & Statistics Is Fun!

The area from 15 to 20 minutes is 5 minutes. Thus: Ch 7 What is the probability of waiting between 15 and 20 minutes? The area from 15 to 20 minutes is 5 minutes. Thus: P(15 < wait time < 20) = height*base = 1 (25-5) *5 = .25 Statistics Are Fun & Statistics Is Fun!

Normal Probability Distribution Is All About Area! Total Area = 1.0 Ch 7 Normal Probability Distribution Is All About Area! Total Area = 1.0 Statistics Are Fun & Statistics Is Fun!

Normal Probability Distribution Formula Ch 7 Normal Probability Distribution Formula Awesome! No, that’s o.k., we can use Appendix or Excel functions! Statistics Are Fun & Statistics Is Fun!

The normal probability distribution is symmetrical about its mean Ch 7 Characteristics of a Normal Probability Distribution (And Accompanying Normal Curve) The normal curve is bell-shaped and has a single peak at the exact center of the distribution The arithmetic mean, median, and mode of the distribution are equal and located at the peak Thus half the area under the curve is above the mean and half is below it The normal probability distribution is symmetrical about its mean If we cut the normal curve vertically at this center value, the two halves will be mirror images Statistics Are Fun & Statistics Is Fun!

The normal probability distribution is asymptotic Ch 7 Characteristics of a Normal Probability Distribution (And Accompanying Normal Curve) The normal probability distribution is asymptotic The curve gets closer and closer to the X-axis but never actually touches it The “tails” of the curve extend indefinitely in both directions The Location of a normal distribution is determined by mean µ The dispersion of a normal distribution is determined by the standard deviation  Now Let’s Look At Some Pictures That Will Show Relationships Amongst Various Means & Standard Deviations Statistics Are Fun & Statistics Is Fun!

Characteristics of a Normal Distribution curve is symmetrical Theoretically, curve extends to infinity Mean, median, and mode are equal There Is A Family Of Normal Probability Distributions Statistics Are Fun & Statistics Is Fun!

Ch 7 Which One Is Normal? Statistics Are Fun & Statistics Is Fun!

Equal Means, Unequal Standard Deviations Ch 7 Equal Means, Unequal Standard Deviations Statistics Are Fun & Statistics Is Fun!

Equal Means, Unequal Standard Deviations Ch 7 Equal Means, Unequal Standard Deviations Statistics Are Fun & Statistics Is Fun!

Equal Means, Unequal Standard Deviations Ch 7 Equal Means, Unequal Standard Deviations Statistics Are Fun & Statistics Is Fun!

Unequal Means, Equal Standard Deviations Ch 7 Unequal Means, Equal Standard Deviations Statistics Are Fun & Statistics Is Fun!

Unequal Means, Unequal Standard Deviations Ch 7 Unequal Means, Unequal Standard Deviations This Family Of Normal Probability Distributions Is Unlimited In Number! Luckily, One Of The Family Members May Be Used In All Circumstances Where The Normal Distribution Is Applicable Standard Normal Distribution Statistics Are Fun & Statistics Is Fun!

Standard Normal Probability Distribution Ch 7 Standard Normal Probability Distribution The standard normal distribution (z distribution ) is a normal distribution with a mean of 0 and a standard deviation of 1 The percentage of area between two z-scores in any normal distribution is the same! Standard deviation & terms may be different, but area will be the same! Normal distributions can be converted to the standard normal distribution using z-values… Statistics Are Fun & Statistics Is Fun!

Define And Calculate z-values Ch 7 Define And Calculate z-values Any normal distribution can be converted, or “standardized” to the standard normal distribution using z-values Z-values: Distance from the mean, measured in units of standard deviation The Formula Is: Z-values are also called: Standard normal value Z score Z statistic Standard normal deviate Normal deviate Remember Your Algebra So That You Can Solve For Any One Of The Variables Statistics Are Fun & Statistics Is Fun!

Standard Normal Probability Distribution Ch 7 Standard Normal Probability Distribution 0 means that there is no deviation from the mean! Statistics Are Fun & Statistics Is Fun!

Convert Value From A Normal Distribution To A Z-score Example 1 Ch 7 Convert Value From A Normal Distribution To A Z-score Example 1 The bi-monthly starting salaries of recent MBA graduates follows the normal distribution with a mean of $2,000 and a standard deviation of $200 What is the z-value for a salary of $2,200? Statistics Are Fun & Statistics Is Fun!

Convert Value From A Normal Distribution To A Z-score, Example 2 Ch 7 Convert Value From A Normal Distribution To A Z-score, Example 2 What is the z-value of $1,700? A z-value of 1 indicates that the value of $2,200 is one standard deviation above the mean of $2,000 (2000 + 1*200) A z-value of –1.50 indicates that $1,700 is 1.5 standard deviation below the mean of $2000 (2000 – 1.5*200) Now we can look at a graph  Statistics Are Fun & Statistics Is Fun!

Ch 7 What is the probability that a foreman’s salary will fall between 1,700 and $2,200? Statistics Are Fun & Statistics Is Fun!

But It Is Really All About Area Under The Curve Remember: Ch 7 But It Is Really All About Area Under The Curve Remember: Areas Under the Normal Curve Empirical Rule (Normal Rule): About 68% of the observations will lie within 1 σ of the mean About 95% of the observations will lie within 2 σ of the mean Virtually all the observations will be within 3 σ of the mean Hints: Many statistical chores can be solved with this normal curve Nevertheless, “The whole world does not fit into a normal curve” Statistics Are Fun & Statistics Is Fun!

Empirical Rule (Normal Rule): Ch 7 Empirical Rule (Normal Rule): Between what two values do about 95% of the values occur? What if you want to find the % of values that lie between z-scores 0 and 1.56? Statistics Are Fun & Statistics Is Fun!

Table In Appendix Or On Inside Cover Ch 7 Table In Appendix Or On Inside Cover Statistics Are Fun & Statistics Is Fun!

Ch 7 Use The Standard Normal Probability Distribution To Find Area: (Table On Inside Back Cover) z  0 1.96 .475 Statistics Are Fun & Statistics Is Fun!

How Can We Use This Standard Normal Curve, Ch 7 Find The Areas z  0 1.96 .475 How Can We Use This Standard Normal Curve, And The Area Under It? Statistics Are Fun & Statistics Is Fun!

Ch 7 Example 1 The daily water usage per person in New Providence, New Jersey is normally distributed Mean = 20 gallons Standard deviation = 5 gallons About 68% of those living in New Providence will use how many gallons of water? +/- 1 standard deviation will give us: About 68% of the daily water usage will lie between 15 and 25 gallons Statistics Are Fun & Statistics Is Fun!

Ch 7 Example 2 What is the probability that a person from New Providence selected at random will use between 20 and 24 gallons per day? Statistics Are Fun & Statistics Is Fun!

Use The Table In The Back Of The Book And Look Up .80 Ch 7 Use The Table In The Back Of The Book And Look Up .80 The area under a normal curve between a z-value of 0 and a z-value of 0.80 is 0.2881 We conclude that 28.81 percent of the residents use between 20 and 24 gallons of water per day See the following diagram: Statistics Are Fun & Statistics Is Fun!

Ch 7 - 5 . 4 3 2 1 x f ( r a l i t b u o n : m = , Area =.2881 “28.81% of the residents use between 20 and 24 gallons of water per day” -4 -3 -2 -1 0 1 2 3 4 z Statistics Are Fun & Statistics Is Fun!

Ch 7 Example 3 What percent of the population use between 18 and 26 gallons per day? Statistics Are Fun & Statistics Is Fun!

Example 3 The area associated with a z-value of –0.40 is .1554 Ch 7 Example 3 The area associated with a z-value of –0.40 is .1554 Because the curve is symmetrical. look up .40 on the right The area associated with a z-value of 1.20 is .3849 .1554 + .3849 = .5403 We conclude that 54.03 percent of the residents use between 18 and 26 gallons of water per day Statistics Are Fun & Statistics Is Fun!

Ch 7 Example 4 Professor Mann has determined that the scores in his statistics course are approximately normally distributed with a mean of 72 and a standard deviation of 5 He announces to the class that the top 15 percent of the scores will earn an A What is the lowest score a student can earn and still receive an A? .50 - .15 = .35  This is the area under the curve! You must look into table and find the value closest to .35 Statistics Are Fun & Statistics Is Fun!

Table In Appendix Or On Inside Cover Ch 7 Table In Appendix Or On Inside Cover Statistics Are Fun & Statistics Is Fun!

Solve for X, The Score You Need To Get An A Ch 7 Solve for X, The Score You Need To Get An A The result is the score that separates students that earned an A from those that earned a B Those with a score of 77.2 or more earn an A Statistics Are Fun & Statistics Is Fun!

Example 5 What is the probability of selecting a shift foreman Ch 7 Example 5 What is the probability of selecting a shift foreman whose salary is between $790 & $1200? Statistics Are Fun & Statistics Is Fun!

Example 5 Find Z-scores Look up area under the curve in the tables Ch 7 Example 5 Find Z-scores Look up area under the curve in the tables Statistics Are Fun & Statistics Is Fun!

Table In Appendix Or On Inside Cover Ch 7 Table In Appendix Or On Inside Cover Statistics Are Fun & Statistics Is Fun!

Ch 7 Example 5 The probability of selecting a shift foreman whose salary is between $790 & $1200 is: .4772 + .4821 = .9593 Statistics Are Fun & Statistics Is Fun!

Example 6 What is the probability of selecting a shift foreman Ch 7 Example 6 What is the probability of selecting a shift foreman whose salary is less than $790? Statistics Are Fun & Statistics Is Fun!

Example 6 Look up the area The area = .4821 .5 - .4821 = .0179 Ch 7 Example 6 Look up the area The area = .4821 .5 - .4821 = .0179 The probability of selecting a shift foreman whose salary is less than $790 is .0179 Statistics Are Fun & Statistics Is Fun!

Ch 7 Summary Statistics Are Fun & Statistics Is Fun!

Ch 7 Finding Area Under The Standard Normal Distribution – It’s All About Area! Statistics Are Fun & Statistics Is Fun!

Ch 7 Finding Area Under The Standard Normal Distribution – Use Formulas & Tables Statistics Are Fun & Statistics Is Fun!

Finding Area Under The Standard Normal Distribution – Four Situations Ch 7 Finding Area Under The Standard Normal Distribution – Four Situations If you wish to find the area between 0 and z (or – z), then you can look up the value directly in the table If you wish to find the area beyond z (or –z), then locate the probability of z in the table and subtract it from .50 If you wish to find the area between two points on different sides of the mean, determine the z-values and add the corresponding areas If you wish to find the area between two points on the same side of the mean, determine the z-values and subtract the smaller area from the larger Statistics Are Fun & Statistics Is Fun!

Ch 7 Use The Normal Distribution To Approximate The Binomial Probability Distribution The normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n Let’s Remember the conditions that must be met before we can run a binomial experiment Statistics Are Fun & Statistics Is Fun!

Ch 7 For An Experiment To Be Binomial It Must Satisfy The Following Conditions: A random variable (x) counts the # of successes in a fixed number of trials (n) Trials make up the experiment Each trail must be independent of the previous trial Outcome of one trial does not affect the outcome of any other trial An outcome on each trial of an experiment is classified into one of two mutually exclusive categories: Success or Failure The probability of success stays the same for each trial (so does the probability of failure) Statistics Are Fun & Statistics Is Fun!

Normal Approximating The Binomial Ch 7 Normal Approximating The Binomial The normal probability distribution is generally a good approximation to the binomial probability distribution when: Statistics Are Fun & Statistics Is Fun!

Mean & Variance of the Binomial Distribution Ch 7 Mean & Variance of the Binomial Distribution Empirical experiments have shown these to be acceptable estimates Statistics Are Fun & Statistics Is Fun!

Continuity Correction Factor Ch 7 Continuity Correction Factor The continuity correction factor of .5 is used to extend the continuous value of x one-half unit in either direction The correction compensates for estimating a discrete distribution by a continuous distribution When you use discrete numbers, you have “gaps” – you need to take an average that will yield a number between We will simply estimate by adding or subtracting the value .5 Statistics Are Fun & Statistics Is Fun!

Continuity Correction Factor Ch 7 Continuity Correction Factor For the probability at least x occur, use the area above (x - .5) For the probability that more than x occur, use the area above (x + .5) For the probability that x or fewer occur, use the area below (x + .5) For the probability that fewer than x occur, use the area below (x - .5) Statistics Are Fun & Statistics Is Fun!

Ch 7 Example A recent study by a marketing research firm showed that 15% of American households owned a video camera. For a sample of 200 homes, What is the probability that less than 40 homes in the sample have video cameras? Step 1: Binomial? Fixed Trails = yes, n = 200 Independent = yes S/F  Success = have video camera, Failure = don’t have  Constant = .15 Statistics Are Fun & Statistics Is Fun!

Ch 7 Example Step 2 : Can we approximate binomial distribution with a standard normal distribution? Step 3 : Calculate μ & σ for the binomial distribution Statistics Are Fun & Statistics Is Fun!

Step 4: Calculate z-value (.5 correction) Ch 7 Step 4: Calculate z-value (.5 correction) What is the probability that less than 40 homes in the sample have video cameras? We use the correction factor, so X is 39.5 The value of z is 1.88 Statistics Are Fun & Statistics Is Fun!

Ch 7 Step 5: Look Up Area From Appendix the area between 0 and 1.88 on the z scale is .4699 So the area to the left of 1.88 is .5000 + .4699 = .9699 The likelihood that less than 40 of the 200 homes have a video camera is about 97% Statistics Are Fun & Statistics Is Fun!

- 5 . 4 3 2 1 f ( x r a l i t b u o n : m = , s2 Ch 7 EXAMPLE 5 Area = .5000+.4699 =.9699 0 1 2 3 4 z=1.88 Statistics Are Fun & Statistics Is Fun!

Ch 7 Summarize Chapter 7 Understand the difference between discrete and continuous distributions Compute the mean and the standard deviation for a uniform distribution Compute probabilities using the uniform distribution List the characteristics of the: Normal probability distribution Standard normal probability distribution Statistics Are Fun & Statistics Is Fun!

Summarize Chapter 7 Define and calculate z values Use the standard normal probability distribution to find area: Above the mean Below the mean Between two values Above one value Below one value Use the normal distribution to approximate the binomial probability distribution Statistics Are Fun & Statistics Is Fun!