5.1 What is Normal?5.1 What is Normal?LEARNING GOAL

Understand what is meant by a normal distribution and be able to identify situations in which a normal distribution is likely to arise.

1. Always plot your data: make a graph2. Look for overall pattern—shape,

outliers, center, spread (SOCS)3. Choose either five number summary

or mean and standard deviation to describe center and spread.

4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by smooth curve.

When exploring a distribution:When exploring a distribution:

What is the shape of this distribution?

Density CurveDensity Curve•This distribution can be approximated by a smooth curve called a density curve.

•This density curve is used in place of histogram to picture the overall shape of the distribution of the data.

•Histograms show the counts or relative frequency of observations in each class by the heights of the bars.

•The density curve shows the proportion of observations in any region by areas under the curve.

•A scale is chosen so that the total area under the density curve is 1.

•What is the shape of this distribution?•The histogram can be approximated by a density curve.

Normal DistributionNormal DistributionSymmetric, bell-shaped distribution

with a single peakIt’s peak corresponds to the mean,

median and mode of the distribution.

It’s variation can be characterized by the standard deviation of the distribution.

The normal distribution is a density curve.

Relative Frequencies and the Relative Frequencies and the Normal DistributionNormal DistributionWith a histogram, the height of a bar

indicates the relative frequency.With a density curve, the area under

the curve corresponding to a range of values on the horizontal axis is the relative frequency of those values.

Because the total relative frequency must be 1, the total area under the density curve must equal 1, or 100%.

•What is the shape of both distributions?•What is the mean? Median? Mode?•What is the area under the curve?•How do the standard deviations compare?

When Can We Expect a When Can We Expect a Normal Distribution?Normal Distribution?Conditions for a Normal DistributionMost data values are clustered near the

mean giving a well-defined single peak.Symmetric distribution with data values

spread evenly around the mean.Tapering tails because larger deviations

from the mean become increasingly rare.

Individual data values result from many factors, such as genetic and environmental factors.

Select the distribution that Select the distribution that appears to be the most normal.appears to be the most normal.

a. b.

c. d.

From each data set, state whether you would From each data set, state whether you would expect it to be normally distributed. Explain expect it to be normally distributed. Explain your reasoning.your reasoning.

Numbers resulting from rolling a single die.

Weights of adult Golden Retriever dogs.

Measured braking reaction times of 18 year old drivers.

ACT scores of all students that took the test in 2008.

5.2 Properties of the 5.2 Properties of the Normal DistributionNormal Distribution

LEARNING GOAL

Know how to interpret the normal distribution in terms of the 68-95-99.7 rule, standard scores, and percentiles.

Is it time to replace your Is it time to replace your TV?TV?Consumer Reports conducted a survey

in which participants were asked how long they owned their last TV set before they replaced it. It was found that the mean time is 8.2 years and the standard deviation is 1.1 years.

Is it reasonable to assume this is a normal distribution?

Is it unusual to replace a TV in 7 years?Is it unusual to have a TV for 14 years

before replacing it?

NotationNotationSampleMean denoted by xStandard deviation denoted by s

PopulationMean denoted by Standard deviation denoted by

Which are parameters? Which are statistics?

68-95-99.7 Rule 68-95-99.7 Rule for a Normal Distributionfor a Normal DistributionAbout 68% of the data points fall within 1 standard deviation of the mean.

About 95% of the data points fall within 2 standard deviations of the mean.

About 99.7% of the data points fall within 3 standard deviations of the mean.

68-95-99.7 Rule68-95-99.7 Rule

+1sd-1sd

68%

+2 sd-2 sd

95%

+3 sd-3 sd

99.7%

Health and Nutrition Examination Health and Nutrition Examination Study of 1976-1980Study of 1976-1980(HANES)(HANES)Heights of adults, aged 18-24

◦womenmean: 65.0 inchesstandard deviation: 2.5 inches

◦menmean: 70.0 inchesstandard deviation: 2.8 inches

Look at examples with this data and the empirical rule.

Unusual valuesUnusual values95% of all values are within 2

standard deviations of the mean, so 5% of all values are more than 2 standard deviations away from the mean.

Unusual values are values that are more than 2 standard deviations away from the mean.

Is it time to replace your Is it time to replace your TV?TV?Consumer Reports conducted a survey

in which participants were asked how long they owned their last TV set before they replaced it. It was found that the mean time is 8.2 years and the standard deviation is 1.1 years. Assuming a normal distribution:

Is it unusual to replace a TV in 7 years?

Is it unusual to have a TV for 14 years before replacing it?

Standard Normal Standard Normal DistributionDistribution

• The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

• The area under the curve is still 1.• Every normal distribution can be transformed

into a standard normal distribution.• To do this we find a standard score with the

following formula:

.

Standard ScoresStandard Scores

z is the standardized score x is the data value is the population mean is the population standard deviation

x

z

The standard score tells us how many standard deviations an observed value is above or below the mean.

ACT scores—2002ACT scores—2002In 2008, the mean composite score in Missouri was

21.4 with a standard deviation of approximately 5.The qualifying score for a Bright Flight scholarship

is 31. What percentile does this represent?The Regents Scholarship at Southeast requires an

ACT score of 27. What percent of students taking the ACT qualify for the Regents Scholarship?

If you wanted to give a scholarship to the top 5% of the scores, what ACT score would you require?

Regular admission (option 2) to Southeast requires an 18 on the ACT and 2.5 high school GPA. What percent of students taking the ACT qualify for admission to Southeast?

American Express CardAmerican Express CardAmerican Express charges merchants higher fees than any

other credit or debit card, according to the USA Today article “American Express fees take flak” (12/23/2004). The company believes they can do this because they claim the customers using the American Express card spend more. The average annual charges per card in 2003 were $9600 according to data from American Express and The Neilson Report. Assume that the annual charges per card are approximately normally distributed with a standard deviation of $2100.

What percent of American Express card users charge less than $4000?

What percent charge more than $16000?What percent charge between $5000 and $10,000

dollars?10% of the card users charge less than what amount?20% of the card users charge more than what amount?