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MTH 126 – FUNDAMENTALS OF STATISTICS

LECTURE NOTES – SECTION 6.2

SECTION 6.2: THE NORMAL MODEL

In the last section we compared discrete and continuous probability distributions. It is much more difficult to find probabilities in continuous probability distributions because we are finding the area under a curve, especially when it’s a curve and not a rectangle.

In this section we will introduce the Normal model and how to find probabilities by finding areas underneath the Normal curve.

The Normal Distribution

The Normal model is the most widely used probability model for continuous numerical variables.

We will use the Normal curve (distribution) for unimodal and symmetric distributions. (Remember symmetric distributions have histograms whose left and right sides are roughly mirror images of one another. Unimodal distributions have histograms with one mound.)

The Normal curve is also sometimes called a Gaussian curve or a bell curve.

Visualizing the Normal Distribution

Normal distributions are unimodal and symmetric distributions, and they can be shown as histograms and/or curves.

Both distributions are nearly symmetric and mound-shaped. The Normal curve is drawn on each histogram.



The Mean and Standard Deviation of a Normal Distribution

The mean of a probability distribution is represented by the Greek character μ (mu). The mean tells us our “balancing point” (center) of the distribution.

The standard deviation of a probability distribution is represented by the Greek character σ (sigma). The standard deviation measures the spread of our distribution by telling us how far away, typically, values are from the mean.

The only way to distinguish among different normal distributions is by their means and standard deviations. We use this to write a short-hand notation to represent a particular Normal distribution. The notation N (μ ,σ ) represents a normal distribution that is centered at the value of μ (the mean), and whose spread is measured by σ (the standard deviation).

(Note: The idea of mean and standard deviation is similar to what we saw in chapter three. The difference was in chapter three we were talking about sets of data, and now we are talking about probability distributions. The Greek characters, μ and σ, are used to avoid confusion with their counterparts for samples of data, x and s.)

Visualizing the Mean and Standard Deviation of a Normal Distribution

Normal distributions are unimodal and symmetric distributions.

Both distributions in the first image have the same mean, but they have different standard deviations.

Both distributions in the second image have the same standard deviations, but they have different means. Also, we can use the notation to describe both distributions. The distribution of women’s heights is N (64,3) and the distribution of men’s heights is N (69,3).



The Empirical Rule - Revisited

The Empirical Rule is a rough guideline for the approximate percentage of data with 1 to 3 standard deviations of the mean.

The Empirical Rule says that if the distribution is unimodal and symmetric, then Approximately 68% of the observations will be within 1 standard deviation of the mean. Approximately 95% of the observations will be within 2 standard deviations of the mean. Nearly all (99.7%) the data will be within 3 standard deviations of the mean.

(We generally consider values to be unusual if they are farther than 2 standard deviations from the mean.)

The following images show the percentages in relation to the mean and standard deviations.

Example 1 – Revisiting the Empirical Rule

According to the Empirical Rule

a. Roughly what percent of z-scores are between −¿1 and 1?

b. Roughly what percent of z-scores are between −¿2 and 2?

c. Roughly what percent of z-scores are between −¿3 and 3?

d. Roughly what percent of z-scores are between 1 and 2?

e. Roughly what percent of z-scores are greater than 0?



Finding Normal Probabilities

When finding probabilities with Normal models, we follow these steps:1) Sketch the Normal curve

2) Find the z-score for the needed number(s): z= x−μσ3) If we are looking for a probability scenario that is

Step 1) Details: Sketch the Normal Curve

Drawing a detailed sketch is the first step to help us to find probabilities using the Normal curve. We want to:

Start with a (bell-shaped) Normal curve. Label the mean and standard deviations. Shade and label the region of interest.

Example 2 – Sketching Normal Curves

The Normal model N (64,3) gives a good approximation of adult women’s heights (in inches) in the United States. Suppose we were to select an adult woman from the United States at random and record her height.

Draw a sketch of the Normal curve to find each probability in question below.

a. What is the probability of randomly selecting a woman taller than 62 inches?(Note: The probability of selecting a woman taller than 62 inches is the same as the probability of selecting a woman 62 inches or taller. We do not have to be too picky about our language when working with continuous variables.)



b. What is the probability of randomly selecting a woman between 62 and 67 inches tall?

c. What is the probability of randomly selecting a woman less than 62 inches tall?

Step 3) Details: Finding Normal Probabilities using the standard Normal model

The standard Normal model, N (0,1), is a Normal model with a mean 0 and standard deviation 1. The standard Normal model allows us to find probabilities for any normal model. The standard normal model is related to z-scores (standard units).

Table 2: Standard Normal Curve Cumulative Probabilities (in the back of your textbook on pages A2-A3) shows the area under the standard Normal curve to the left of z. (A small portion of the table is shown below.)

Reading Table 2: Standard Normal Curve Cumulative Probabilities

The numbers along the left margin, when joined to the numbers across the top represent z-scores.

The circled number in the table on the next page represents the area under the curve and to the left of 1.00 standard unit, visually shown in the image on the right.



Finding Normal Probabilities

When finding probabilities with Normal models, we follow these steps:1) Sketch the Normal curve

o Label it appropriatelyo Shade in the region of interest

2) Find the z-score for the needed number(s): z= x−μσ3) If we are looking for a probability scenario that is

o “less than a value” – look up the probability for the z-score in the standard Normal Curve Cumulative Probabilities Table. This is your probability.

o “between two values” – look up the probability for both z-scores in the standard Normal Curve Cumulative Probabilities Table. Find the difference of the probabilities.

o “greater than a value” – look up the probability for the z-score in the Standard Normal Curve Cumulative Probabilities Table. Subtract this from 1.



Example 3 – Women’s Height (less-than probability)

Using the Normal model N (64,3), suppose we were to select an adult woman from the United States at random and record her height.

What is the probability of randomly selecting a woman less than 62 inches tall?

1) Sketch the curve.

2) Find the z-score.

z= x−μσ

=¿

3) Determine the probability.

We will use this row.



Example 4 – Women’s Height (between probability)


What is the probability of randomly selecting a woman between 62 and 67 inches tall?


2) Find the z-scores.

z1=x−μσ

=¿

z2=x−μσ

=¿


Example 5 – Women’s Height (greater-than probability)


What is the probability of randomly selecting a woman taller than 62 inches?



z= x−μσ

=¿




Example 6 – Small Seal Pups

The normal model N (29.5,1 .2) gives a good approximation of the distribution of Pacific harbor seal newborn pups’ lengths (in inches). Suppose we were to select a seal pup at random and record its weight.

What is the probability of randomly selecting a seal pup that is shorter than 28.0 inches?



z= x−μσ

=¿


Example 7 – A Range of Seal Pups


What is the probability of randomly selecting a seal pup that is between 27 and 31 inches long?


2) Find the z-scores.

z1=x−μσ

=¿

z2=x−μσ

=¿




STATCRUNCH – FINDING NORMAL PROBABILITIES (The TI-84 instructions are on page 301.)

- Go to “Stat”- Select “Calculator”- Select “Normal”- Enter the “Mean” and “Standard Deviation” values at the bottom of the window. - Select the “standard” or “between” at the top of the window.

o For “standard” select ≤ or ≥ and enter the single value in box next to the inequality symbol.

o For “between” enter values on both sides of the inequality symbols.- “Compute”- Selecting the “68-95-99.7 ticks” button at the top of the window will label the horizontal axis

using the mean and standard deviation. (Note: It does not need to be re-selected more than once.)

Example 6 – Small Seal Pups – Revisited


What is the probability of randomly selecting a seal pup that is shorter than 28.0 inches?


2) Use StatCrunch to determine the probability.



Example 7 – A Range of Seal Pups – Revisited


What is the probability of randomly selecting a seal pup that is between 27 and 31 inches long?



Example 8 – Females’ SAT Scores

The normal model N (500,100) gives a good approximation of the distribution of quantitative SAT scores for college-bound female high school seniors in 2012.

a. What percentage of female college-bound students had scores below 400?






b. What percentage of female college-bound students had scores above 650?




c. What percentage of female college-bound students had scores between 450 and 600?



Finding Measurements from Percentiles (Finding Inverse Normal Values)

When finding measurements from percentiles:

1) Sketch the Normal curveo Label it appropriatelyo Shade in the region of interest

2) Find the z-score from the percentile



A percentile is a value on a scale of 100 that indicates the percent of a distribution that is equal to or below it

(For example, the 20th percentile is the value below which 20% of the observations may be found.)

3) Convert the z-score to proper units: x=μ+zσ

Example 9 – Inverse Normal or Normal?

Suppose that the amount of money people keep in their online PayPal account follows a Normal model.

For each situation, identify whether the question asks for a measurement or a Normal probability.

a. A PayPal employee wonders what the probability is that a randomly selected customer will have less than $150 in his/her account.

b. A PayPal customer wonders how much money he would have to put in the account to be in the 90th percentile.

Example 10 – Females’ SAT Percentiles

The Normal model N (500,100) gives a good approximation of the distribution of quantitative SAT scores for college-bound female high school seniors in 2012.

A scholarship is awarded to a student who scores at the 96th percentile or above. What is that score?



z=¿

3) Convert the z-score to units:

x=μ+zσ=¿



Example 11 – Females’ SAT Scores IQR


Answer the following questions about the interquartile range (IQR) of the distribution of scores.

a. Find the SAT Score at the 25th percentile and find the SAT Score at the 75th percentile.


2) Find the z-score for the 25th and 75th percentile (from the table).

z25=¿

z75=¿

3) Convert the z-scores to units to find the SAT scores at the 25th and 75th percentiles.:

x=μ+z25 σ=¿

x=μ+z75 σ=¿

b. Find the interquartile range (IQR) for the SAT scores.

c. Is the IQR larger or smaller than one standard deviation? Explain



STATCRUNCH – FINDING THE INVERSE NORMAL (The TI-84 instructions are on page 301.)

- Go to “Stat”- Select “Calculator”- Select “Normal”- Enter the “Mean” and “Standard Deviation” values at the bottom of the window. - Select the “standard” at the top of the window.

o For “standard” select ≤ and enter the single value in the box after the equal sign.- “Compute”- Selecting the “68-95-99.7 ticks” button at the top of the window will label the horizontal axis

using the mean and standard deviation. (Note: It does not need to be re-selected more than once.)

Example 10 – Females’ SAT Percentiles – Revisited


A scholarship is awarded to a student who scores at the 96th percentile or above. What is that score?


2) Use StatCrunch to determine the score.



Example 8 – Females’ SAT Scores

According to the National Health Center, the heights of 6-year-old girls are normally distributed with a mean of 45 inches and a standard deviation of 2 inches.

a. In which percentile is a 6-year-old girl who is 46.5 inches tall?



According to the National Health Center, the heights of 6-year-old girls are normally distributed with a mean of 45 inches and a standard deviation of 2 inches.

b. If a 6-year-old girl who is 46.5 inches tall grows up to be a woman at the same percentile of height, what height will she be? Assume women are distributed as N (64,2.5 ).


2) Use StatCrunch to determine the height.



Snapshot: The Normal ModelWhat is it? A model of distribution for some numerical variables.What does it do? Provides us with a model of the distributions of probabilities for many

real-life numerical variables.How does it do it? The probabilities are represented by the area underneath the bell-

shaped curve.How is it used? If the normal model is appropriate, it can be used for finding

probabilities or for finding measurements associated with particular percentiles.

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