Normal Distribution
Consider the following: Four coins are flipped and the number of heads is recorded.
1. List all of the possible outcomes. Are all outcomes equally likely? Explain.
2. What is the theoretical probability of getting 4 heads? 3 heads? 2 heads? 1 head? 0 heads?
Calculate the percent likelihood.
3. What observations can be made about the theoretical probabilities?
4. On the axis below, complete the histogram of the probability of each number of heads.
a. Draw a point at the midpoint of the top of each bar.
Connect the points in a smooth curve.
b. What do you notice about the graph’s shape?
c. What do you observe about the graph’s mean,
median, and mode?
Characteristics of a normal curve:
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Normal Distribution
Normally distributed data sets have a normal curve graph.
 Examples of normally distributed data: heights, tread life of tires, SAT scores
Normal curves follow the 68-95-99.7% Rule (aka Empirical Rule or 3 Sigma Rule)
Consider the following: Pulse rates are normally distributed with a mean of 72 and a standard
deviation of 12. Label the normal model below.
The 68-95-99.7% Rule tells us that
 68% of all people will have pulse rates within 1 standard deviation of the mean.
That means that 68% of people have pulse rates between __________ and __________.
 95% of all people will have pulse rates within 2 standard deviations of the mean.
That means that 95 % of people have pulse rates between __________ and __________.
 99.7% of all people will have pulse rates within 3 standard deviations of the mean.
That means that 99.7 % of people have pulse rates between __________ and __________.
Values outside of _____________ or the __________ range are considered unusual.
5. Given a normally distributed data set of 500 observations measuring tree heights in a forest,
what is the approximate number of observations that fall within two standard deviations from
the mean?
6. A normally distributed data set containing the number of ball bearings produced during a
specified interval of time has a mean of 150 and a standard deviation of 10. What percentage
of the observed values fall between 140 and 160?
Normal Distribution
Finding the area under a normal curve means to find the probability of a value being greater than
the rest of the data.
Areas can be found under a normal
curve by using the 68-95-99.7 rule if the
areas are bounded at places where an
exact deviation occurs.
Areas that are not bounded at ±1, ±2, or
±3 standard deviations can be found
using a z-table.
Consider the following: A corn chip factory packs chips in bags with normally distributed
weights with a mean of 12.4 oz. and a standard deviation of 0.14 oz.
7. On the graph to the right, label the mean and three
standard deviations above and below the mean.
8. Shade the region that indicates the percentage of bags
that contain less than 12.64 oz.
9. Calculate the z-score of 12.64.
10. Use the z-table (Standard Normal Probabilities Table) to find the area associated with the zscore and interpret your result.
a. The probability that a data value will fall ___________ the data value associated with a
z-score of __________ is ______________ .
b. 12.64 falls in the ________ percentile. This means that _______ percent of the data in
the distribution fall ____________ the value associated with a z-score of ___________.
c. The probability that a value from the data set will fall __________ this value is
Normal Distribution
11. On the graph at the right, label and shade the region
that represents the likelihood a bag will contain between
12.1 and 12.76 oz.
12. Calculate the z-score for each and find the probability.
13. How can we use these values to determine the probability of choosing a bag between 12.1
and 12.76 oz.?
Consider the following: The length of rhinoceros’ horns are normally distributed with a mean of
10.1 cm and a standard deviation of 1.4 cm.
14. On the graph to the right, label the mean and three
standard deviations above and below the mean.
15. What percent of rhinos have horns shorter than
10.1 cm? _______ Longer than 10.1cm? _______
16. Calculate the probability of a rhinos’ horn being longer than 9 cm. Interpret your results.
17. Find P  9.5  x  10.5