Normally Distributed Variables

1
Unit 11

• Normally Distributed Variables

2
Discrete vs Continuous

• There are two distinctly different types of
  numeric variables
• A binomial variable is said to be a discrete
  variable since it can only assume whole number
  values, The number of successes is limited to 0,
  1, 2, 3, etc.
• A continuous variable can take on any value
  within a specified range of values

3
Examples

• Some examples of continuous variables include
• The number of hours of sleep a college student
  gets on a week night
• The height of college women
• The speed of vehicles traveling on a certain
  stretch of highway

4
Its Your Turn

• Think of a few examples of continuous variables

5
Normally Distributed Variables

• Many continuous variables have what is known as a
  normal distribution
• These include height, weight, blood pressure, SAT
  scores, sleep times and many others.
• Normally distributed variables follow the
  empirical rule

6
The 68-95-99.7 Rule

• The Empirical Rule requires that a continuous
  variable must
• Have a symmetrical distribution
• Have 68 of the population within one standard
  deviation of the mean
• Have 95 of the observations within two standard
  deviations of the mean
• Have 99.7 of the observations within three
  standard deviations of the mean

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Example

• Suppose that it takes you 20 minutes, on average,
  to drive to campus each day with a standard
  deviation of 2 minuets. Suppose also that a
  normal model approximates the distribution of
  times.
• How often will you arrive on campus in less than
  22 minutes?
• How often will it take you more than 24 minutes?

8
Probability it takes less than 22 minutes 68
16 84
9
Probability it takes more than 24 minutes is 2.5
10
Areas Under the Curve

• The total area under a normal curve is 1
• The area under specific portions of the normal
  curve correspond to the probability that the
  variable will be in that specific range.

11

• For college women pulse rates are normally
  distributed with a mean of 76 and a standard
  deviation of 8. Which graph represents the
  proportion of women,
• greater than 86
• less than 66

12

• The number of hours of sleep for college students
  on weeknights is approximately normally
  distributed with a mean of 7 and a standard
  deviation of 1.7. Which graph represents the
  proportion of students who sleep
• Less than 7 hours?
• Between 5 and 9 hours?
• What does graph B represent?

13
The Secret

• The secret to finding the area under a normal
  curve is measuring how many standard deviations a
  value is from the mean.
• The quantity
• is a measure of how many standard deviations
  an observation, x, is from the mean.
• It is called a standardized score.

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Example

• For women in their twenties pulse rates are
  normally distributed with a mean of 76 and a
  standard deviation of 8.
• What is the standardized score for a woman with a
  pulse rate of 88?
• Interpret this z score

15
Solution

• A woman with a pulse rate of 88 has a pulse
  rate that is 1.5 standard deviations above the
  mean

16
Standard Normal Curve

• Standardized z scores are normally distributed
  with
• and

17
Questions

• On the standard normal curve
• What percent of the observations are between -1
  and 1?
• What percent of the observations are between -2
  and 2?
• What percent of the observations are between -3
  and 3?

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Standard Normal Calculator

• To find areas under the standard normal curve the
  easy way use the following on-line calculator
• Standard Normal Calculator
• The area above 1.5 is .0668 so 6.68 of women in
  their twenties have a pulse rate of 88 or more

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Pulse Rates Again

• For women in their twenties pulse rates are
  normally distributed
• To find the probability a woman has a pulse rate
  of 88 or more we need to find the area above 1.5
  on the standard normal curve

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Pregnancy Durations

• The distribution of the duration of human
  pregnancies is approximately normal with a mean
  of 270 days and a standard deviation of 15 days.
  What proportion of pregnancies come to term in
• Less than 244 days?
• More than 275 days?
• Between 260 and 280 days?

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Less Than 244 Days

• Area .0418
• 4.18 of pregnancies
• come to term in less
• than 244 days

22
More Than 275 Days

• Area is .3707
• 37.1 of pregnancies
• Take more than 275
• Days to come to term

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Between 260 and 280 Days

• Area .4972
• Probability 1 2(.2514) .4972
• 49.7 of pregnancies
• come to term in between
• 260 and 280 days

24
Candy Bars Yum!

• The weights of individual Milky Way bars vary
  slightly but are approximately normally
  distributed with a mean of 2.2 ounces and a
  standard deviation of .04 ounces.
• The wrapper list the weight as 2.13 ounces. What
  percent of the candy bars weight less than
  advertised?
• What percent of the candy bars weight between
  2.16 and 2.27 ounces?

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Candy Bar Solution

• z -.175, area to left of -1.75 is .0401
• 4.01 of the candy bars weigh less than what it
  says on the wrapper.
• Area between z -1 and z 1.75 is
• 1 (.1587 .0401) .8012
• 80.12 of the candy bars weigh between 2.16 and
  2.27 ounces.

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Using Your TI-83/84

• To find the area between two z-scores under the
  standard normal curve
• Choose normalcdf from DISTRbutions menu
• Enter normalcdf(zLeft, zRight)
• To find area to the right of z 2 enter
  normalcdf(2, 99)
• To find the area to the left of z 2 enter
  normalcdf(-99, 2)

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Examples

• Find the area to the right of z 1.75
• Normalcdf(1.75, 99) .0401
• Find the area between z -1.2 and z 2.1
• Normalcdf(-1.2, 2.1) .8670

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Your Turn

• Find the area below z -2.22
• Find the area in between z 1.23 and
  z 2.79
• Area below z -2.22 is .0132
• Area between z 1.23 and z 2.79 is .1067

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Pregnancy and Percentiles

• Recall that for duration of pregnancy the mean is
  270 days and the standard deviation is 15 days
• How many days does a woman have to be pregnant to
  be in the top 10 percentile?

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Use TechnologyWork the Process in Reverse

• Use the normal calculator and enter the area to
  the right of z as .10. The calculator reports a
  z-score of 1.282
• x 1.282(15) 270 289 days
• 10 of all pregnancies last longer than 289 days

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Underweight Candy Bars

• If the weight of candy bars is normally
  distributed with a mean of 2.2 ounces and a
  standard deviation of .04 ounces how much does a
  candy bar have to weight to be in the bottom 15
  of candy bar weights?

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Solution

• area .15
• z -1.036
• -1.036(.04) 2.2 2.16
• 15 of the candy bars weigh less than 2.16 ounces

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On the TI-83/84 Calculator

• To find the z-score that marks the 90th
  percentile
• Choose invNorm( from the DISTR menu
• Enter the command invNorm(.90)
• invNorm(.90) 1.28155 or 1.28