Download Statistics – Chapter 5 – Normal Distributions Section 5.1 – Intro to

Statistics – Chapter 5 – Normal Distributions
Section 5.1 – Intro to Normal Distributions
Properties of a normal distribution:
 Bell-shaped distribution
 Symmetric distribution with mean = median = mode
 Area under curve = 1 (50% of the probability on each side of mean)
 Curve approaches (but never touches) x-axis on each side of the mean
 Concave down between -1 and +1 and concave up beyond one standard deviation
from the mean…inflection points at -1 and +1.
Empirical Rule (68-95-99.7): In any normal distribution the probabilities follow this
pattern:
Section 5.2 – The Standard Normal Distribution
The Standard Normal is a normal with mean = 0 and the standard deviation = 1.
Points on the x-axis are z-scores. These z-scores correspond to cumulative probabilities (or
areas) under the standard normal curve. The table in our statistics book gives the cumulative
probability (area) to the left of the given z-score.
Examples:
Probability less than a z-score: Find area to the left.
P(z<1.30) = .9032
Probability greater than a z-score: Find area to the right:
P(z>1.30) = 1 - .9032 = .0968
Statistics – Chapter 5 – Normal Distributions
Probability between z-scores: Find the area between.
P(.21< z<1.80) = .3809
P(z<1.80) – P(z <.21) = .9641 – .5832 = .3809
Probability outside z-scores: Add the areas outside.
P(z<.21 or z>1.80) = 1 - .3809 = .6191
P(z<.21) + P(z>1.80) = .5832 + .0359 = .6191
Section 5.3 – Finding Probabilities from Normal Distributions
Use z 
x 

where  is the mean of the distribution,  is the standard deviation and x is the
data value for the probability of interest. Once the z-score is determined, the standard normal
table can be used to determine the probability.

Probability on the calculator:
2nd Vars (Distr)
2: normalcdf
normalcdf(x low, x high, , )
Example: Given a distribution with mean  = 45 and standard deviation  = 12, what is the
probability of getting a data value of 39 or more.
P(x > 39) ….
z
39  45
 .5
12
P(z > -.5) = 1 – P(z < -.5) = 1 - .3085 = .6915
calculator: normalcdf(39, 10000, 45, 12) = .6915

(Note: 10,000 was selected as x high because it is high enough to represent “all”
probability above 39.)
Statistics – Chapter 5 – Normal Distributions
Section 5.4 – Finding Data Values from Normal Distributions (given probabilities)
It is also possible to find a data value related to a given probability. From the probability, use
the table to determine the z-score. From the z-score, solve for the data value…x =  + z.
Data Value on the calculator:
2nd Vars (Distr)
3: invNorm
invNorm(probability to the left of the data value, , )
Example: Given a distribution with mean  = 45 and standard deviation  = 12, what is the
data value for P25 (or the 25th percentile).
The z-score related to the 25th percentile is -.67.
x = 45 + (-.67)12 = 37.0
For this distribution, 25% of data values will be less than 37.0 and 75% will be greater.
calculator: invNorm(.25, 45, 12) = 37.0
Section 5.5 – Using the Central Limit Theorem and finding Probabilities for Averages
Often we want to find probabilities related to averages. For example, what’s the probability
that the average height of a sample of 20 adult females is > 65 inches.
To do this we need the mean and standard deviation of the sampling distribution.
mean of the sampling distribution = mean of the original distribution: xÝÝÝ 
standard error (or standard deviation) of the sampling distribution = standard deviation
of the original distribution divided by the square root of the sample
 size: xÝÝÝ 

n
Example: What’s the probability that the average height of a sample of 20 adult females > 65
inches. Heights of females are normally distributed with mean 64 and standard dev. 2.75.

xÝÝÝ 64 and  xÝÝÝ
2.75
 .61
20
z
65  64
1.63
.61
P(z>1.63) = 1 – .9484 = .0516
if either of the following criterion are met:
 Note: Thisprocess can only be used

 n<30: If sample size is less than 30 then the data must come from a normal
distribution, or,
 n30: Sample size is greater than or equal to 30. In this case it doesn’t matter what the
original distribution is since the Central Limit Theorem says that averages from that
distribution will be normally distributed (for sufficiently large sample sizes).