Math 116 and Math 116A– Study Guide for Chapters 5 and 6 = AGRESTI
Chapter 5 – ONLY STUDY THE CLASS’ NOTES – In the book there are many other topics that we
don’t cover in class
WE USED THE EXPERIMENT OF ROLLING A 1-6 DIE TO cover the following topics:
1) Sample space
2) Outcomes
3) Events
4) Probability rules
5) Complement rule
6) Simulation of the experiment using the calculator (randInt(1,6))
7) Experimental probability versus theoretical probability
8) Law of Large Numbers
9) Constructing a probability distribution and a probability histogram
10) Finding the mean and the standard deviation of a probability distribution with the calculator
Section 6.1 – Summarizing outcomes and their probabilities
1. Random variable (covered in chapter 5 handout)
2. Probability distribution of a discrete random variable (covered in chapter 5 handout)
3. Probability histogram (covered in chapter 5 handout)
4. Mean and standard deviation of a probability distribution (with the calculator) - (covered in chapter 5
handout)
5. Weighted average
6. Probability distribution of a continuous random variable
7. From histograms to density curves
8. Characteristics of a density curve
9. Area, probability, proportion
10. Uniform distribution
11. Probabilities = area under a density curve
Section 6.2 – The Normal Distribution
12. Characteristics of a normal distribution
a) Complete description of a normal distribution with the mean and the standard deviation
b) Parameters of a normal distribution – notation used
c) How does the change of the mean (mu) and the standard deviation (sigma) affect a normal
distribution
13. Standard normal distribution (the z-score distribution: mean = 0 and standard deviation = 1)
14. Finding probabilities
o With procedure and table (sketch graph, label, shade the requested probability, find the zscore, use the table to find the area to the left of the z-score, answer the question)
o With calculator (use Normalcdf(left, right, mean, standard deviation)
15. Finding scores
o With procedure and table (sketch graph, label, identify the area to the left of the score that
you are looking for, use table to find the z-score by reading the table from inside out, use the
formula to find the score: x = mu + z*sigma)
o With calculator (use InvNorm(area to the left, mean, standard deviation)
Now you know how the approximate percentages of the empirical rule were obtained: 68 – 95 – 99.7


The 68-95-99.7 rule for exactly normal distributions
For a given normal distribution: N(mu, sigma), show a sketch and label the mean and the scores that
are 1, 2, and 3 standard deviations away from the mean. Use the graph to answer questions of the
type:
o 68% of the scores are between ______ and ______
o 95% of the scores are between ______ and ______
o 99.7% of the scores are between ______ and ______
o Give the % (area/probability/proportion) between two given scores
o Give the % (area/probability/proportion) below a certain score
o Give the % (area/probability/proportion) above a certain score
Summary
We can sometimes describe the overall pattern of a distribution by a density curve. A density curve has total area
1 underneath it. An area under a density curve gives the proportion of observations that fall in a range of values.

A density curve is an idealized description of the overall pattern of a distribution that smooths out the
irregularities in the actual data. We write the mean of a density curve as μ and the standard deviation of a
density curve as σ to distinguish them from the mean and standard deviation s of the actual data.

The mean, the median, and the quartiles of a density curve can be located by eye. The mean μ is the
balance point of the curve. The median divides the area under the curve in half. The quartiles and the median
divide the area under the curve into quarters. The standard deviation σ cannot be located by eye on most
density curves.

The mean and median are equal for symmetric density curves. The mean of a skewed curve is located
farther toward the long tail than is the median.

The Normal distributions are described by a special family of bell-shaped, symmetric density curves,
called Normal curves. The mean μ and standard deviation σ completely specify a Normal distribution N(μ,
σ). The mean is the center of the curve, and σ is the distance from μ to the change-of-curvature points on either
side.

To standardize any observation x, subtract the mean of the distribution and then divide by the standard
deviation. The resulting z-score
says how many standard deviations x lies from the distribution mean.

All Normal distributions are the same when measurements are transformed to the standardized scale. In
particular, all Normal distributions satisfy the 68 − 95 − 99.7 rule, which describes what percent of
observations lie within one, two, and three standard deviations of the mean.
If x has the N(μ, σ) distribution, then the standardized variable z = (x − μ)/σ has the standard Normal
distribution N(0, 1) with mean 0 and standard deviation 1. Table A gives the cumulative proportions of
standard Normal observations that are less than z for many values of z. By standardizing, we can use Table A for any
Normal distribution.