Section 2.2 Normal Distributions

Mrs. DanielAP Statistics

Section 2.2Normal Distributions

After this section, you should be able to…

DESCRIBE and APPLY the 68-95-99.7 Rule

DESCRIBE the standard Normal Distribution

PERFORM Normal distribution calculations

ASSESS Normality

Normal Distributions• All Normal curves are symmetric, single-peaked, and bell-shaped

• A Specific Normal curve is described by giving its mean µ and standard deviation σ.

Two Normal curves, showing the mean µ and standard deviation σ.

Normal Distributions

• We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).• Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. • The mean of a Normal distribution is the center of the symmetric Normal curve. • The standard deviation is the distance from the center to the change-of-curvature points on either side.

Normal Distributions are Useful…• Normal distributions are good descriptions for some

distributions of real data.

• Normal distributions are good approximations of the results of many kinds of chance outcomes.

• Many statistical inference procedures are based on Normal distributions.

Normal Distributions

will appear AGAIN

and AGAIN.

Chapter 6, Chapter 8,

Chapter 9 and

Chapter 10!

Although there are many different sizes and shapes of Normal curves, they all have properties in common.

The 68-95-99.7 Rule

The 68-95-99.7 Rule (“The Empirical 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.

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55) and the range is between 0 and 12.

a) Sketch the Normal density curve for this distribution.

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55) and the range is between 0 and 12.

a) Sketch the Normal density curve for this distribution.

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).

b) Using the Empirical Rule, what percent of ITBS vocabulary scores are less than 3.74?

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).

b) Using the Empirical Rule, what percent of ITBS vocabulary scores are less than 3.74?

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).?

c) Using the Empirical Rule, what percent of the scores are between 5.29 and 9.94?

The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).?

c) Using the Empirical Rule, What percent of the scores are between 5.29 and 9.94?

Importance of Standardizing

• There are infinitely many different Normal distributions; all with unique standard deviations and means.

• In order to more effectively compare different Normal distributions we “standardize”.

• Standardizing allows us to compare apples to apples.

• We can compare SAT and ACT scores by standardizing.

The Standardized Normal DistributionAll Normal distributions are the same if we measure in units of size σ from the mean µ as center.

The standardized Normal distribution is the Normal distribution with mean 0

and standard deviation 1.

1. Draw and label an Normal curve with the mean and standard deviation.

2. Calculate the z- scorex= variable µ= meanσ= standard deviation

3. Determine the p-value by looking up the z-score in the Standard Normal table.

4. Conclude in context.

How to Standardize a Variable:

The Standard Normal TableBecause all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.

The Standard Normal table is a table of the areas under the standard normal curve. The table entry for each value “z” is area under the curve to the LEFT of z.

The area to left is called the “p-value”

• Probability

• Percent

Using the Standard Normal TableRow: Ones and tenths digitsColumn: Hundredths digit

Practice: What is the p-value for a z-score of -2.33?

Using the Standard Normal Table

Using the Standard Normal Table, find the following:

Z-Score P-value

-2.23

1.65

.52

.79

.23

Let’s PracticeIn the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour (mph) on his first serves. Assume that the distribution of his first serve speeds is Normal with a mean of 115 mph and a standard deviation of 6.2 mph. About what proportion of his first serves would you expect to be less than 120 mph? Greater than?

1. Draw and label an Normal curve with the mean and standard deviation.

2. Calculate the z- scorex= variable µ= meanσ= standard deviation

3. Determine the p-value by looking up the z-score in the Standard Normal table.

Z .00 .01 .02

0.7 .7580 .7611 .7642

0.8 .7881 .7910 .7939

0.9 .8159 .8186 .8212

P(z < 0.81) = .7910

4. Conclude in context.

We expect that 79.1% of Nadal’s first serves will be less than 120 mph.

We expect that 20.9% of Nadal’s first serves will be greater than 120 mps.

Let’s Practice

When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?

When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?

Step 1: Draw Distribution

Step 2: Z- Scores

When x = 325, z =325 - 304

82.63

When x = 305, z =305 - 304

80.13

Using Table A, we can find the area to the left of z=2.63 and the area to the left of z=0.13.0.9957 – 0.5517 = 0.4440.

Step 3: P-values

About 44% of Tiger’s drives travel between 305 and 325 yards.

Step 4: Conclude In Context

Normal Calculations on CalculatorCalculates Example

NormalCDF Probability of obtaining a value BETWEEN two values

What percent of students scored between 70 and 95 on the test?

NormalPDF Probability of obtaining PRECISELY or EXACTLY a specific x-value

What is the probability that Suzy scored a 75 on the test?

InvNorm X-value given probability or percentile

Tommy scored a 92 on the test; what proportion of students did he score better than?

TI-Nspire: NormalCDFNormalcdf- “Area under the curve between two points”

1.Select Calculator (on home screen), press center button. 2.Press menu, press enter.3.Select 6: Statistics, press enter. 4.Select 5: Distributions, press enter. 5.Select 2: Normal Cdf, press enter. 6.Enter the following information:

1. Lower: (the lower bound of the region OR 1^-99)2. Upper: (the upper band of the region OR 1,000,000)3. µ: (mean)4. Ơ: (standard deviation)

7.Press enter, number that appears is the p-value

TI-Nspire: NormalPDFNormalpdf- “Exact percentile/probability of a specific event occurring”

1.Select Calculator (on home screen), press center button. 2.Press menu, press enter.3.Select 6: Statistics, press enter. 4.Select 5: Distributions, press enter. 5.Select 1: Normal Pdf press enter. 6.Enter the following information:

1. Xvalue (not a percent)2. µ: (mean)3. Ơ: (standard deviation)

7.Press enter, number that appears is the p-value

TI-Nspire: InvNorminvNorm- “Exact x-value at which something occurred”

1.Select Calculator (on home screen), press center button. 2.Press menu, press enter.3.Select 6: Statistics, press enter. 4.Select 5: Distributions, press enter. 5.Select 3: Inverse Norm press enter. 6.Enter the following information:

1. Area (enter as a decimal)2. µ: (mean)3. Ơ: (standard deviation)

7.Press enter, number that appears is the p-value

When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?

When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?

Suzy bombed her recent AP Stats exam; she scored at the 25th percentile. The class average was a 170 with a standard deviation of 30. Assuming the scores are normally distributed, what score did Suzy earn of the exam?

Suzy bombed her recent AP Stats exam; she scored at the 25th percentile. The class average was a 170 with a standard deviation of 30. Assuming the scores are normally distributed, what score did Suzy earn of the exam?

When Can I Use Normal Calculations?!

• Whenever the distribution is Normal.

• Ways to Assess Normality: – Plot the data.

• Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped.

– Check whether the data follow the 68-95-99.7 rule.

– Construct a Normal probability plot.

Normal Probability Plot

• These plots are constructed by plotting each observation in a data set against its corresponding percentile’s z-score.

Interpreting Normal Probability Plot• If the points on a Normal probability plot lie close to a

straight line, the plot indicates that the data are Normal.

• Systematic deviations from a straight line indicate a non-Normal distribution.

• Outliers appear as points that are far away from the overall pattern of the plot.

Summary: Normal Distributions• The Normal Distributions are described by a special family of bell-shaped,

symmetric density curves called Normal curves. The mean µ and standard deviation σ completely specify a Normal distribution N(µ,σ). The mean is the center of the curve, and σ is the distance from µ to the change-of-curvature points on either side.

• All Normal distributions obey the 68-95-99.7 Rule, which describes what percent of observations lie within one, two, and three standard deviations of the mean.

• All Normal distributions are the same when measurements are standardized. The standard Normal distribution has mean µ=0 and standard deviation σ=1.

• Table A gives percentiles for the standard Normal curve. By standardizing, we can use Table A to determine the percentile for a given z-score or the z-score corresponding to a given percentile in any Normal distribution.

• To assess Normality for a given set of data, we first observe its shape. We then check how well the data fits the 68-95-99.7 rule.

Additional Help

Finding Areas Under the Standard Normal Curve

Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81.

Step 1: Look up area to the left of 0.81 using table A.

Step 2: Find the area to the left of -1.25

Finding Areas Under the Standard Normal Curve

Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81.

Step 3: Subtract.