Chapter 2.2

STANDARD NORMAL DISTRIBUTIONS

Normal Distributions

• Last class we looked at a particular type of density curve called a Normal distribution.

• All Normal distributions are described by two parameters: ________ and __________

• Because of this, we can abbreviate a Normal distribution as ____(___, ___)

• Another important quality of Normal distributions is that the follow the __________ rule. This rule states that ____% of the data falls within 1 standard deviation of the mean, ____% falls within 2 standard deviations and _____% falls within 3 standard deviations.

μ σ

μ σN

Empirical68

9599.7

The Standard Normal Distribution• All normal distributions are the same if we measure in units of

size σ about the mean μ as center. • Changing these units requires that we standardize (like we did

in 2.1)

σZ = x - μ

• If the variable we standardize has a normal distribution, then so does the new variable, z

• The new distribution is called the standard Normal Distribution

Great…but why is that useful?• Remember, the area under a density curve is a

proportion of the observations in a distribution. – The area under the entire density curve is ____. – The proportion of observations to the left of the

median is_____.

1

.5

• We can find the proportion of observation that lie within any range of values simply by finding the area under the curve.

The standard Normal Table

• Because standardizing Normal distributions makes them all the same, we can use a single table to find the areas under a Normal distribution.

• This table is called the standard Normal table. – It’s inside the front cover of you textbook!– You will be given this table on the AP exam

CAREFUL!!!!

The standard Normal Table

• Example: Find the proportion of observations from the standard Normal distribution that are less than -2.15.

• For the value of z = -2.15, the area is 0.0158

Using the standard Normal table…

• Caution: the area that we found was to the LEFT of z = -2.15. In this case, that is what we were looking for.

• HOWEVER if the problem had asked for the area lying to the right of -2.15. What would that answer be?

Area to the Right

• The total area under the curve is _____. • So if 0.0158 lies to the left of -2.15… • Then _____ - 0.0158= _______ lies to the right of -2.15.

1

1 0.9842

How do you avoid making a mistake when asked to find the area to the RIGHT?

• Always sketch the Normal curve, mark the z-value, and shade the area of interest (aka the area you are looking for in the problem)

• THEN, when you get you answer, CHECK TO SEE IF IT IS REASONABLE!!!

Practice

• Exercise 2.29

Putting it all Together:Solving Problems Involving Normal Distributions• Step 1: State the problem in terms of the

observed variable x. Draw a picture of the distribution and shade the area of interest.– Hint…use σ and μ

• Step 2: Standardize and draw a picture. We need to standardize x to restate the problem in terms of a standard Normal variable z. Draw a new picture to show the area of interest under our now standard Normal curve.

Putting it all Together:Solving Problems Involving Normal Distributions• Step 3: Use the table. Find the are under the

standard Normal curve using Table A. (careful if the problem asks for the area to the right)

• Step 4: Conclusion. Write your conclusion in the context of the problem. – Just saying “the area under the curve that is less

that 2.1” means nothing! Your results should tell you something about the data.

Example: Cholesterol and Young Boys• For 14-year-old boys, the mean is μ = 170 milligrams of cholesterol per deciliter of blood

(mg/dl) and the standard deviation σ = 30 mg/dl. • Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys

have more than 240 mg/dl of cholesterol?

Finding a Value when Given a Proportion• What if you wanted to know what score you would have to get

in order to place among the top 10% of your class on a test?

• Sometimes, we may be asked to find the observed value with a given proportion of the observations above or below it.

• To do this, we just read Table A going backwards. In other words, find the proportion you are looking for in the body of the table, figure out the corresponding z-score, and then “unstandardize” to get the observed value.

Inverse Normal Calculation Example• Scores on the SAT Verbal test in recent years

follow approximately the N(505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

Inverse Normal Calculation Example• Scores on the SAT Verbal test in recent years follow approximately the

N(505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

Practice!

• 2.31a, b• 2.32

Assessing Normality

• The normal distribution provides a good model for some distributions of real data.

• However, not all distributions are Normal. • It is important to assess the Normality of

distributions before we assume that they are normal.

• This will be very important when we learn about statistical inference procedures (much later)

Assessing Normality Method 1

• One method for assessing normality is to construct a histogram or a stemplot and then see if the graph is approximately bell-shaped and symmetric about the mean.

• Histograms and stemplots can reveal important “non-Normal” features of a distributions such as skewness, outliers, or gaps and clusters.

Method 1 Continued• For example, this distribution of vocabulary scores

appears Normal. – The distribution is bell-shaped, it is roughly symmetric,

there are no gaps or clusters, and there do not appear to be any outliers.

Method 2

• MEAN = 6.8585• STDEV = 1.5952

2.07 3.67 5.26 6.86 8.45 10.05 11.64 x - 3s x - 2s x – s x x + s x + 2s x+3s

1 21 129 331 318 125 21 1

We can improve the effectiveness of our plots by marking x, x ± s, x ± 2s on the horizontal axis. Then compare the counts of observations in each interval using the empirical rule.

Method 2 Continued

2.07 3.67 5.26 6.86 8.45 10.05 11.64 x - 3s x - 2s x – s x x + s x + 2s x+3s

1 21 129 331 318 125 21 1

Method 2 Continued…

• Because the actual counts of our distribution follow the empirical rule very closely, we can confirm that the Normal distribution with μ = 6.86 and σ = 1.595 fits the data well.

Method #2 for Assessing Normality• Construct a normal probability plot. This requires the

use of your graphing calculator. • Basically…without a calculator..– Arrange the observed data values from smallest to largest.

Record what percentile of the data set each value occupies. (i.e., the smallest observation in a set of 20 is the 5% point, the second smallest is the 10%, etc)

– Use the standard Normal distribution table to find the z-scores for these percentiles. (i.e., z = -1.645 is the 5% point of the standard Normal distribution)

– Plot each data point x against the corresponding z.

Method 2 Continued

• Let’s interpret some Normal probability plots!

Normal Probability Plots• If you draw a

line, it appears that most of the data lies close to a straight line.

• HOWEVER, the points above and below the line represent outliers in our data.

The only substantial deviations from the line are short horizontal runs of points.These represent repeated observations of the same value. The phenomenon is called granularity and does not effect Normality.

Normal Probability Plots• This is the Normal

probability plot for guinea pig survival times.

• Draw a line through the leftmost points (smallest observations)

• Notice that the larger observations fall systematically ABOVE the line.– In other words, the right-

of-center observations have larger values than the Normal distribution

– Therefore, the distribution is right skewed

Normal Probability Plots

• The Normal probability plot indicates that the data is left-skewed because the smallest observations fall below the line.

Interpreting Normal Probability Plot

Graphing Normal Probability Plots on your Calculator

• Enter the test scores for Mr. Pryor’s statistics class on page 116 into L1 on your calculator.

• Press 2nd , Y= (STAT PLOT)• Turn Plot 1 ON• Select the type on the lower right• Data List: L1• Data Axis: x• Mark (doesn’t matter)• Press Zoom, 9:ZoomStat• You should have a probability plot!

Using your calculator: Finding Areas with normalcdf

• You can find the areas under the Normal curve using normalcdf. • For 14-year-old boys, the mean is μ = 170 milligrams of cholesterol per deciliter of

blood (mg/dl) and the standard deviation σ = 30 mg/dl. • Levels above 240 mg/dl may require medical attention. What percent of 14-year-

old boys have more than 240 mg/dl of cholesterol?

Using Your Calculator: invNorm• Finally, we can use our calculators to calculate raw or

standardized values given the area under the Normal curve or a relative frequency.

• Scores on the SAT Verbal test in recent years follow approximately the N(505, 110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

Next time on Statistics AP

• The Chapter 2 Test will be on TUESDAY!– It will cover all of Chapter 2

• SO on Friday we will– Review Chapter 2

• YOUR HOMEWORK:– Exercises 2.31(c), 2.33, 2.34, 2.36, 2.40, 2.48– YOU MAY USE YOUR CALCULATOR BUT YOU STILL

NEED THE SKETCH AND Z-SCORE