Chapter 3 The Normal Distributions. Chapter outline 1. Density curves 2. Normal distributions 3. The...
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Transcript of Chapter 3 The Normal Distributions. Chapter outline 1. Density curves 2. Normal distributions 3. The...
Chapter 3Chapter 3
• The Normal Distributions
Chapter outlineChapter outline• 1. Density curves• 2. Normal distributions• 3. The 68-95-99.7 rule• 4. The standard normal distribution• 5. Normal distribution calculations - 1:
proportion?• 6. Normal distribution calculations - 2: z-
score?
Density curvesDensity curves
• A density curve is a curve that – 1. is always on or above the x-axis– 2. Has area exactly 1 underneath it.Special Case : Normal curve
A density curve describes the overall pattern of a distribution. Areas under the density curve represent proportions of the total number of observations.
Density curvesDensity curves
Density curvesDensity curves
Density curvesDensity curves
Density curvesDensity curves
• Properties of density curve:– Median of a density curve: the equal-area
point the point that divides the area under the curve in half.
– Mean of a density curve: the balance point, at which the curve would balance if made of solid material.
• Notation: mean ( ), standard deviation ( ), for a density curve.
Density curvesDensity curves
Normal distributionsNormal distributions
• Possible values vary from• Notation: • A density curve -
– It is single peaked and bell-shaped.– It never hits x-axis. It is above x-axis.– Centered at . That is, determines the location of
center.– Having spread around the mean
to),( N
Figure 3.7 (P.62) Two normal curves, showing Figure 3.7 (P.62) Two normal curves, showing the mean and standard deviationthe mean and standard deviation
The 68-95-99.7 ruleThe 68-95-99.7 rule
• For :• 1. 68% of the observations fall within of• 2. 95% of the observations fall within 2 of
• 3. 99.7% of the observations fall within 3 of
),( N
The 68-95-99.7 ruleThe 68-95-99.7 rule
The 68-95-99.7 ruleThe 68-95-99.7 rule
• Example 3.2 (P.63)
The standard normal distributionThe standard normal distribution
• Mean=0, standard deviation =1
• Notation:
• If x follows , follows
)1,0(N
),( N
x )1,0(N
The standard normal distributionThe standard normal distribution
• Example 3.3 (P.65)
• Example 3.4 (P.66)
How to use Table AHow to use Table A
• To find a proportion: start with values on edges and find a value within the table
• To find a z-score: start in the middle of table and read the edges.
Normal distribution calculations 1: proportion?Normal distribution calculations 1: proportion?
• By using Table A: areas under the curve of N(0,1) are provided.– 1. State in terms of – 2. State the problem in terms of x– 3. Standardize x in terms of z– 4. Draw a picture to show the area we are interested in – 5. Use Table A to find the required area
• Area to the left?• Area to the right?• Area in between?
),( N
Normal distribution calculations 1: proportion?Normal distribution calculations 1: proportion?
• Example 3.5 (P.68)
• Example 3.6 (P.69)
• Example 3.7 (P.70)
Normal distribution calculations 2: z-scores?Normal distribution calculations 2: z-scores?
• So far, we find a proportion using specific value(s) on x-axis.
• Question: What if proportion is given and we want to find the specific value(s) on x-axis that give(s) given proportion?– 1. State in terms of – 2. State the problem in terms of z– 3. Use Table A– 4. Unstandardize from z to x (if needed)
Normal distribution calculations 1: proportion?Normal distribution calculations 1: proportion?
• Exercise 3.10 (P.70 )
• Exercise 3.20 (P. 75)
Normal distribution calculations 2: z-scores?Normal distribution calculations 2: z-scores?
• Example 3.8 (P.72)
• Exercise 3.12 (P.73)