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)