Transcript Normal Distribution for Video

Introductory Statistics
Outline of Lesson 05 Videos
1. Density
Curve
2. 68-95-99.7 Rule
3. Z-Score
4. Converting Z-Score to Probability
5. Percentiles for Normal Distribution
6. QQ-Plots – Assessing Normality
Density Curve
The total area under the curve equals 1.
2. The density curve always lies on or above
the horizontal axis.
1.
Area inside curve
equals 1.
Horizontal Axis
Outline of Lesson 05 Videos
1. Density
Curve
2. 68-95-99.7 Rule
3. Z-Score
4. Converting Z-Score to Probability
5. Percentiles for Normal Distribution
6. QQ-Plots – Assessing Normality
68-95-99.7 Rule
The empirical rule:
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Approx. 68% of all of the data fall within 1 standard deviation of the mean
Approx. 95% of all of the data fall within 2 standard deviations of the mean
Approx. 99.7% of all of the data fall within 3 standard deviations of the mean
Mean - (# of Std Deviations * σ) = Lower Value
Mean + (# of Std Deviations * σ) = Upper Value
Sketching the normal distribution is strongly recommended.
The mean incubation time of fertilized chicken eggs kept at a certain
temperature is 21 day. Incubation times are normally distributed with a
standard deviation (σ) of one day. According to the Empirical Rule,
approximately 95% of incubations will have times between which two values?
21 +- 2 (1) = (19,23)
68-95-99.7 Rule
68-95-99.7 Rule
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The council for Graduate Medical Education found that medical residents’
mean number of hours worked in a week is 81.7 The number of hours
worked per week is normally distributed with a standard deviation of 6.9
hours.
According to the Empirical Rule, approximately 95% of students will have
work times per week between which two values?
95% of Data will fall within: 81.7 +- 2*(6.9) = (67.9,95.5)
What about 99.7% of students?
99.7% of Data will fall within: 81.7 +- 3*(6.9) = (61,102.4)
What approximate percent of data will fall between 74.8 hours and 88.6
hours?
Both 74.8 and 88.6 are 1 standard deviation from the mean so 68% of the
data will fall within those two points
68-95-99.7 Rule
Outline of Lesson 05 Videos
1. Density
Curve
2. 68-95-99.7 Rule
3. Z-Score
4. Converting Z-Score to Probability
5. Percentiles for Normal Distribution
6. QQ-Plots – Assessing Normality
Standard Normal Distribution
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Suppose the variable X is normally distributed with mean µ
and standard deviation σ. Then the random variable Z is
normally distributed with mean 0 and standard deviation of 1.
The Z-Score is calculated as follows:
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Z-Score represents the number of standard deviations a value (X) is
away from the mean (µ).
Uses for the Z-Score
1. Determine values that may be extreme
2. Compare values from two different types of normal
distributions
3. Being capable of using normal probabilities to calculate the
probabilities of specific events
Determining Extreme Values
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The mean incubation time of fertilized chicken eggs kept at a certain
temperature is 21 day. Incubation times are normally distributed with a
standard deviation of one day. What is the Z-Score of an incubation time of
20.5 days? Would that be considered extreme? How about 25 days?
Z=20.5-21/1 = -0.5 – Not too extreme
Z=25-21/1 = 4 Appears to be extreme
The council for Graduate Medical Education found that medical residents’
mean number of hours worked in a week is 81.7 The number of hours worked
per week is normally distributed with a standard deviation of 6.9 hours. What
is the Z-Score of a study time in a week of 84 hours? Would that be
considered extreme? How about 110 hours?
Z=(84-81.7)/6.9 = 0.33 - Not too extreme
Z=(110-81.7)/6.9 = 4.10 - Appears to be extreme
Comparing Scores across different distributions
Player
Batting Avg Mean
Std. Dev. Z-Score
3.788
Ted Williams
0.406
0.281
0.033
4.111
Rod Carew
0.388
0.277
0.027
3.964
George Brett
0.39
0.279
0.028
Tony Gwynn
0.394
0.293
0.032
3.156
Who had the best baseball season?
Outline of Lesson 05 Videos
1. Density
Curve
2. 68-95-99.7 Rule
3. Z-Score
4. Converting Z-Score to Probability
5. Percentiles for Normal Distribution
6. QQ-Plots – Assessing Normality
Steps to Finding Area or Probability Under
Normal Curve
1.
2.
Convert (standardize) the values of X using the
standard normal variable Z (Find the Z-Score)
Find the area under the standard normal curve by
using the applet
a. http://byuimath.com/apps/normprob.html
Steps to Finding Area or Probability Under
Normal Curve
The length of human pregnancies from conception to birth is normally
distributed with mean 266 Days and a standard deviation of 16 days.
What is the probability of a pregnancy lasts less than 240 day?
1. Convert (standardize) the values of X using the standard normal
variable Z (Find the Z-Score)
Z = (240 – 266) / 16 = -1.625
2.
Find the area under the standard normal curve (Using the applet)
0.052
Steps to Finding Area or Probability Under
Normal Curve
The length of human pregnancies from conception to birth is normally
distributed with mean 266 Days and a standard deviation of 16 days.
What is the probability of a pregnancy lasts more than 270 days?
1. Convert (standardize) the values of X using the standard normal
variable Z (Find the Z-Score)
Z = (270 – 266) / 16 = 0.25
2.
Find the area under the standard normal curve (Using the applet)
0.401
Steps to Finding Area or Probability Under
Normal Curve
The length of human pregnancies from conception to birth is normally
distributed with mean 266 Days and a standard deviation of 16 days.
What is the probability that a pregnancy lasts between 241 and 291 days?
1. Convert (standardize) the values of X using the standard normal variable Z
(Find the Z-Score)
Z = (241 – 266) / 16 = -1.563
Z = (291 – 266) / 16 = 1.563
2.
Find the area under the standard normal curve (Using the applet)
0.882
Outline of Lesson 05 Videos
1. Density
Curve
2. 68-95-99.7 Rule
3. Z-Score
4. Converting Z-Score to Probability
5. Percentiles for Normal Distribution
6. QQ-Plots – Assessing Normality
Steps to Finding a value when given the area
or percentile (less than 50th Percentile)
1.
If needed, click on the areas with the
normal distribution curve so that the
only area shaded in blue is on the left
side of the left line
2.
Type in the Percentile number in
decimal form in the area box. (e.g. 40th
Percentile)
3.
Use the Z score under the left red line
and put into the formula X = µ + (Z*σ)
to find the value of interest (X) which is
the percentile you are looking for (e.g.
µ = 266 Days and σ = 16 days for
human pregnancies.
X = µ + (Z*σ) = 266 + (-.253 * 16)
= 261.95
Steps to Finding a value when given the area
or percentile (greater than 50th Percentile)
1.
If needed, click on the areas with the
normal distribution curve so that the
areas shaded in blue are on the left
side of the left line and in between
both red lines
2.
Type in the Percentile number in
decimal form in the area box. (e.g. 90th
Percentile)
3.
Use the Z score under the right red
line and put into the formula X = µ +
(Z*σ) to find the value of interest (X)
which is the percentile you are looking
for (e.g. µ = 266 Days and σ = 16 days
for human pregnancies.
X = µ + (Z*σ) = 266 + (1.28 * 16) =
286.48
Outline of Lesson 05 Videos
1. Density
Curve
2. 68-95-99.7 Rule
3. Z-Score
4. Converting Z-Score to Probability
5. Percentiles for Normal Distribution
6. QQ-Plots – Assessing Normality
QQ Plot
Normally Distributed
Not Normally Distributed