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Chapter 6 Important Information
Section 6.2
1. Normal Distribution: If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, as in the
figure below, we can say that it has a normal distribution.
Normal Distribution 
Uniform Distribution 
Area = 1
2. Uniform Distribution: A continuous random variable has a uniform distribution if its values are spread evenly over the range of
possibilities. The graph of a uniform distribution results in a rectangular shape.
3. Requirements for a Density Curve: A density curve is the graph of a continuous probability distribution. It must satisfy the
following properties:
1. The total area under the curve must equal 1.
2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)
*Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.
4. Standard Normal Distribution: a normal probability distribution with µ = 0 and σ = 1. The total area under its density curve is
equal to 1.
5. The standard normal distribution has three properties:
1. It’s graph is bell-shaped.
2. It’s mean is equal to 0 ( = 0).
3. It’s standard deviation is equal to 1 ( = 1).
6. Method for Finding Normal Distribution Areas
(When being asked to find an area or probability you will use the same method)
In the following notations, z represents a z score, which is a standardized value.
P(a < z < b) : denotes the probability that the z score is between a and b.
To find this probability in your calculator, type: normalcdf(a, b, µ, σ)
a
b
P(z > a) : denotes the probability that the z score is greater than a.
To find this probability in your calculator, type: normalcdf(a, 99999, µ, σ)
a
P(z < a) : denotes the probability that the z score is less than a.
To find this probability in your calculator, type: normalcdf(–99999, a, µ, σ)
7. Method for Finding z scores when given an Area or Probability
a
1. Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the
z score(s) being sought.
2. Determine the area to the left and enter the following into your calculator: InvNorm(area to the left, µ, σ)
3. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem.
8. 𝒁𝜶 Notation: the expression zα denotes the z score with an area of α to its right.
To find this z score in your calculator, type: invNorm(1 – α, µ, σ)
Section 6.3
9. Procedure for Finding Areas with a Nonstandard Normal Distribution:
1. Sketch a normal curve, label the mean and specific x values, then shade the region representing the desired probability.
2. Use the normalcdf feature on your calculator to find the area of the shaded region. Refer back to the notation in number 6 on the
previous page to find the desired area/probability.
In the following notations, x represents a non-standardized individual value.
P(a < x < b) : denotes the probability that the z score is between a and b.
To find this probability in your calculator, type: normalcdf(a, b, µ, σ)
a
b
P(x > a) : denotes the probability that the z score is greater than a.
To find this probability in your calculator, type: normalcdf(a, 99999, µ, σ)
a
P(x < a) : denotes the probability that the z score is less than a.
To find this probability in your calculator, type: normalcdf(–99999, a, µ, σ)
a
10. Helpful Hints:
1. Choose the correct (right/left) side of the graph. A value separating the top 10% from the others will be located on the right
side of the graph, but a value separating the bottom 10% will be located on the left side of the graph.
2. A z score must be negative whenever it is located in the left half of the normal distribution.
3. Areas (or probabilities) are positive or zero values, but they are never negative.
11. Procedure for Finding Values Using the InvNorm Feature on Your Calculator
1. Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the
x value(s) being sought.
2. Determine the area to the left and enter the following into your calculator: InvNorm(area to the left, µ, σ)
3. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem.
Section 6.4
12. The sampling distribution of the mean is the distribution of sample means, with all samples having the same sample size n taken
from the same population. (The sampling distribution of the mean is typically represented as a dot plot or histogram.)
13. The sampling distribution of the proportion is the distribution of sample proportions, with all samples having the same sample
size n taken from the same population.
14. Unbiased Estimators: Sample means, variances, and proportions are unbiased estimators, because they target the population
parameter. (They have a mean that is equal to the mean of the corresponding population.) These statistics are better in estimating the
population parameter.
15. Biased Estimators: Sample medians, ranges and standard deviations are biased estimators, that is they do NOT target the
population parameter.
Section 6.5
16. Central Limit Theorem
When you are Given these two pieces…
1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation σ.
2. Simple random samples all of size n are selected from the population. (The samples are selected so that all possible samples
of the same size n have the same chance of being selected.)
…you can make these Conclusions:
1. The distribution of sample x will, as the sample size increases, approach a normal distribution.
2. The mean of the sample means is the population mean µ.
3. The standard deviation of all sample means is

n
17. Central Limit Theorem Notation
P(a < 𝒙 < b): denotes the probability that the z score is between a and b.
𝝈
*To find this probability in your calculator, type: normalcdf(a, b, µ, )
√𝒏
P(𝒙 > a): denotes the probability that the z score is greater than a.
*To find this probability in your calculator, type: normalcdf(a, 999999999, µ,
P(𝒙 < a): denotes the probability that the z score is less than a.
*To find this probability in your calculator, type: normalcdf(–99999999, a, µ,
𝝈
√𝒏
𝝈
√𝒏
)
)