CHAPTER 3: The Normal Distributions Basic Practice of Statistics - 3rd Edition CHAPTER 3: The Normal Distributions Basic Practice of Statistics 7th Edition Lecture PowerPoint Slides Chapter 5

Spreadsheet Assignment 3 Let’s take a quick look…

Questions about assignments, course, etc.? Questions about today’s reading (3.0-3.3)

Entry Slip We learn in Chapter 3 that there are infinitely many different normal distributions. However, we can know exactly which normal distribution we’re talking about if we know two bits of information. What are they?

What would it look like if we rolled a die 1,000 times?

Exploring a distribution Plot your data. Look for the overall pattern (shape, center, and variability) and for striking deviations such as outliers. Calculate summaries for center and variability. Now we add one more step to this strategy: Check whether the data can be represented by a density curve.

Density Curve Area = 1.00 = 100%

Density Curves The mean and standard deviation computed from actual observations (data) are denoted by x and s, respectively. The mean and standard deviation of the actual distribution represented by the density curve are denoted by µ (“mu”) and σ (“sigma”), respectively.

Guinea pigs

Figure 3.3, The Basic Practice of Statistics, © 2015 W. H. Freeman Guineau pigs injected with tuberculosis bacteria Figure 3.3, The Basic Practice of Statistics, © 2015 W. H. Freeman

Congenital Heart Disease

Our Breath Data

Normal Distributions One important class of density curves are the Normal curves, which describe Normal distributions. All Normal curves are symmetric, single-peaked, and bell-shaped Any specific Normal curve is described by giving its mean µ and standard deviation σ.

Normal Density Curve Example: Here is a histogram of vocabulary scores of 947 seventh graders. The smooth curve drawn over the histogram is a mathematical model for the distribution.

The 68-95-99.7 Rule The 68-95-99.7 Rule In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.

Questions? Next due date: Wednesday Chapter 0-3 Exam: Friday

What did our breath data look like the second time?

Question Which test score is better? A 20 on the ACT or a 900 on the SAT? We need more info. ACT (µ = 21, σ = 5) SAT (µ = 1000, σ = 200) Scores on each test follow an approximately Normal distribution.

Get a visual! ACT (µ = 21, σ = 5) SAT (µ = 1000, σ = 200) Which test score is better? A 20 on the ACT or a 900 on the SAT?

We can plot both points on the same distribution. ACT (µ = 21, σ = 5) SAT (µ = 1000, σ = 200) Which test score is better? A 20 on the ACT or a 900 on the SAT?

The Standard Normal Distribution The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.

Z-score The number of standard deviations above the mean. 𝑧= x − µ σ 𝑧= 20 −21 5 = - .2 𝑧= 900 − 1000 200 = - .5 The number of standard deviations above the mean. Called a “standardized score.” Can be used with any distribution, but is most common (and useful) with the Standard Normal distribution.

Why are SAT scores important?

Question Which is more exceptional? The fact that ‘s pulse is _____ or the fact that can hold his breath for _______ seconds? We need more info. Women’s pulse (µ = 78, σ = 11) College holding breath (µ = 44, σ = 18) Scores on each test follow an approximately Normal distribution. Two random students; 1 exceptional thing for each; google the mean and SD

College holding breath Get a visual! Women’s pulse (µ = 78, σ = 11) College holding breath (µ = 44, σ = 18)

Cumulative Proportions The cumulative proportion for a value x in a distribution is the proportion of observations in the distribution that are less than or equal to x.

Normal Calculations Using Table A to find Normal proportions Step 1. State the problem in terms of the observed variable x. Draw a picture that shows the proportion you want in terms of cumulative proportions. Step 2. Standardize x to restate the problem in terms of a standard Normal variable z. Step 3. Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve.

Cumulative Proportions—Example Example: Who qualifies for college sports? The combined scores of the almost 1.7 million high school seniors taking the SAT in 2013 were approximately Normal with mean 1011 and standard deviation 216. What percent of high school seniors meet this SAT requirement of a combined score of 820 or better? Find the z-score of an 820 in N(1011, 216). 𝑧= 820 −1011 216 = - .88

Cumulative Proportions—Example Convert z-score of -0.88 to a proportion. =NORMDIST or Table A Here is the calculation in a picture: So about 81% qualified for college sports.

Starters Exam format and what to study Questions on Moodle Quiz 3? Comments on SA3 Questions from Chapters 0-2 More practice with Chapter 3

Using the symmetry of the Standard Normal Table What percent of the standard normal distribution is below a z-score of -1.5? (Note: z-score and z-value are the same thing.) .0668 = 6.68% What percent of the standard normal distribution is above a z-score of 1.5? 1 - .9332 = .0668 = 6.68% What generalization can you make?

Using the symmetry of the Standard Normal Table Using the left-side of Table A, find the percentage of the standard Normal distribution that is… Above a z-score of 1.2. = below a z-score of -1.2 = .1151 = 11.51% Below a z-score of 0.4. = above a z-score of -0.4 = (1 - .3446) = .6554 = 65.54% Above a z-score of -.1.73. = (1 - .0418) = .9582 = 95.82%

Finding a slice of the standard Normal distribution What percent of the standard Normal distribution is between a z-score of -0.8 and a z-score of 1.3?

What percent of the standard Normal distribution is between a z-score of -0.8 and a z-score of 1.3? 21.19% is below a z-score of -0.8. 90.32% is below a z-score of 1.3. So, 90.32% - 21.19% = 69.13% is in between the values.

“Backward” Normal Calculations Using Table A Given a Normal proportion Step 1. State the problem in terms of the given proportion. Draw a picture that shows the Normal value, x, you want in relation to the cumulative proportion. Step 2. Use Table A, the fact that the total area under the curve is 1, and the given area under the standard Normal curve to find the corresponding z-value. Step 3. Unstandardize z to solve the problem in terms of a non-standard Normal variable x.

Basic Practice of Statistics - 3rd Edition Finding a Value Given a Proportion SAT reading scores for a recent year are distributed according to a N(504, 111) distribution. How high must a student score in order to be in the top 10% of the distribution? Chapter 5

Basic Practice of Statistics - 3rd Edition Normal Calculations SAT reading scores for a recent year are distributed according to a N(504, 111) distribution. How high must a student score in order to be in the top 10% of the distribution? In order to use table A, equivalently, what score has cumulative proportion 0.90 below it? .10 504 ? N(504, 111) .90 Chapter 5

Basic Practice of Statistics - 3rd Edition Normal Calculations .10 504 ? How high must a student score in order to be in the top 10% of the distribution? Look up the closest probability (closest to 0.90) in the table. Find the corresponding standardized score. The value you seek is that many standard deviations from the mean. z .07 .08 .09 1.1 .8790 .8810 .8830 1.2 .8980 .8997 .8015 1.3 .8147 .8162 .8177 z = 1.28 Chapter 5

Basic Practice of Statistics - 3rd Edition Normal Calculations .10 504 ? How high must a student score in order to be in the top 10% of the distribution? z = 1.28 We need to “unstandardize” the z-score to find the observed value (x): x = 504 + z(111) = 504 + [(1.28 )  (111)] = 504 + (142.08) = 646.08 A student would have to score at least 646.08 to be in the top 10% of the distribution of SAT reading scores for this particular year. Chapter 5

Or, on Google Sheets, use =NORMINV(.9,504,111)