Section 2.2 Normal Distributions Mrs. Daniel AP Statistics

Section 2.2 Normal Distributions After this section, you should be able to… DESCRIBE and APPLY the 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 somedistributions of real data. Normal distributions are good approximations of the resultsof 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 Normalcurves, they all have properties in common. The Rule The 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 7 th 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 7 th 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 7 th 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 7 th 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 7 th 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 7 th 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 Normaldistributions; all with unique standard deviationsand means. In order to more effectively compare different Normal distributions we “standardize”. Standardizing allows us to compare apples toapples. We can compare SAT and ACT scores bystandardizing.

The Standardized Normal Distribution All Normal distributions are the same if we measure inunits 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- score x= variableµ= mean σ= 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 Table Because all Normal distributions are the same when westandardize, we can find areas under any Normal curvefrom a single table.The Standard Normal table is a table of the areas underthe 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 Table Row: Ones and tenths digitsColumn: Hundredths digitPractice: What is the p-value for a z-score of ?

Using the Standard Normal Table Using the Standard Normal Table, find thefollowing: Z-ScoreP-value

Let’s Practice In the 2008 Wimbledon tennis tournament,Rafael Nadal averaged 115 miles per hour (mph)on his first serves. Assume that the distributionof his first serve speeds is Normal with a mean of115 mph and a standard deviation of 6.2 mph.About what proportion of his first serves wouldyou expect to be less than 120 mph? Greaterthan?

1. Draw and label an Normal curve with the mean andstandard deviation.2. Calculate the z- scorex= variableµ= meanσ= standard deviation

3. Determine the p-value by looking up the z-score inthe Standard Normal table. Z P(z < 0.81) =.7910

4. Conclude in context.We expect that 79.1% of Nadal’s first serves will be lessthan 120 mph.We expect that 20.9% of Nadal’s first serves will be greaterthan 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?

Step 1: Draw Distribution Step 2: Z- Scores

Using Table A, we can find the area to the left of z=2.63 and the area to the left of z= – = Step 3: P-values About 44% of Tiger’s drives travel between 305 and 325 yards. Step 4: Conclude In Context

Normal Calculations on Calculator CalculatesExample NormalCDFProbability of obtaining a value BETWEEN two values What percent of students scored between 70 and 95 on the test? NormalPDFProbability of obtaining PRECISELY or EXACTLY a specific x-value What is the probability that Suzy scored a 75 on the test? InvNormX-value given probability or percentile Tommy scored a 92 on the test; what proportion of students did he score better than?

TI-Nspire: NormalCDF Normalcdf - “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: NormalPDF Normalpdf - “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: InvNorm invNorm - “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 distancethe ball travels can be described by N(304, 8).What percent of Tiger’s drives travel between305 and 325 yards?

Suzy bombed her recent AP Stats exam; shescored at the 25 th 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 thegraph is approximately symmetric and bell-shaped. – Check whether the data follow the rule. – Construct a Normal probability plot.

Normal Probability Plot These plots are constructed by plotting each observationin 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 theoverall 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 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 rule.

Additional Help

Finding Areas Under the Standard Normal Curve Find the proportion of observations from the standard Normal distribution that are between and 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 StandardNormal Curve Find the proportion of observations from the standard Normal distribution that are between and Step 3: Subtract.