Download Let`s Be Normal Learning Task

Unit 7: Task 4
Name ________________________________
Let’s Be Normal Learning Task
Until this task, we have focused on distributions of discrete data. We will now direct
our attention to continuous data. Where a discrete variable has a finite number of
possible values, a continuous variable can assume all values in a given interval of
values. Therefore, a continuous random variable can assume an infinite number of
values.
We will focus our attention specifically on continuous random variables with
distributions that are approximately normal. Remember from last semester, that
normal distributions are symmetric, bell-shaped curves that follow the Empirical Rule.
The Empirical Rule for Normal Distributions states that
68% of the data values will fall within one
standard deviation of the mean,
95% of the data values will fall within two
standard deviations of the mean, and
99.7% of the data values will fall within three standard deviations of the mean.
Dotplots were used to explore this type of distribution and you spent time determining
whether or not a given distribution was approximately normal. In this task, we will
use probability histograms and approximate these histograms to a smooth curve that
displays the shape of the distribution without the lumpiness of the histogram. We will
also assume that all of the data we use is approximately normally distributed.
1. This is an example of a probability histogram of a continuous random variable
with an approximate mean of 5 and standard deviation of 1. A normal curve
with the same mean and standard deviation has been overlaid on the histogram.
a. What do you notice about these two
graphs?
2. First, you must realize that normally distributed data may have any value for
its mean and standard deviation. Below are two graphs with three sets of
normal curves.
Unit 7: Task 4
Name ________________________________
a. In the first set, all three curves have the same mean, 10, but different standard
deviations. Approximate the standard deviation of each of the three curves.
Tallest Curve: __________
Middle Curve:___________
Smallest Curve:_________
b. In this set, each curve has the same standard deviation. Determine the standard
deviation and then each mean.
Left Curve:
  ______   ________
Middle Curve:
Right Curve:
  ______   ________
  ______   ________
c. Based on these two examples, make a conjecture regarding the effect changing the
mean has on a normal distribution. Make a conjecture regarding the effect changing
the standard deviation has on a normal distribution.
Unit 7: Task 4
Name ________________________________
The Standard Normal Distribution is a normal distribution with a mean of 0 and a
standard deviation of 1. To more easily compute the probability of a particular
observation given a normally distributed variable, we can transform any normal
distribution to this standard normal distribution using the following formula:
z
X 

When you find this value for a given value, it is referred to as the z-score. The z-score is
a standard score for a data value that indicates the number of standard deviations that
the data value is away from its respective mean.
3. Use the formula and definition for z-score to calculate the z-score for the
mean of a variable.
4. Looking at the standard normal curve below, you see that half of the area of
the curve lies below the mean of the distribution, therefore P( X   )  0.5 .
Another way to express this probability is P( z  0)  0.5 .
a. Now let’s say that you want to know the probability of an observation falling below
15 for a variable that is normally distributed with a mean of 15 and standard deviation
of 3.
Unit 7: Task 4
Name ________________________________
5. We can use the z-score to find the probability of many other events. Let’s
explore those now.
a. Suppose that the mean time a typical American teenager spends doing homework
each week is 4.2 hours. Assume the standard deviation is 0.9 hour. Assuming the
variable is normally distributed, find the percentage of American teenagers who spend
less than 3.5 hours doing homework each week.
First, find the z-score for X = 3.5. Now, use the method you teacher has shown you to
determine the probability of P(X < 3.5). Sketch a normal curve for this situation and
shade the probability you have just found on the curve.
b. The average height of adult American males is 69.2 inches. If the standard deviation
is 3.1 inches, determine the probability that a randomly selected adult American male
will be at most 71 inches tall. Assume a normal distribution.
6. We can also use z-scores to find the percentage or probability of events above
a given observation.
Due to the symmetry of the curve,
We can use the Empirical Rule to
we can quickly deduce that the
determine that the probability of
probability of a value occurring above
a value happening one or more
the mean is 0.50.
standard deviations above the mean
is 0.16, therefore .
P( z  1)  0.16
Unit 7: Task 4
Name ________________________________
We can also compute P ( z  a ) using the Complement Rule that you first learned in
Accelerated Mathematics I.
P( z  a)  1  P( z  a )
Use this rule to compute the following probabilities.
a. The average on the most recent test Mrs. Cox gave her Physics students was 73 with
a standard deviation of 8.2. Assume the test scores were normally distributed.
Determine the probability that a student in Mrs. Cox’s class scored an A on the test.
height of at least 64 inches.
7. We can also determine the probability between two values of a random
variable.
P(2  z  1)  P( z  1)  P( z  2)
P(1.5  z  0.75)  ___________
Unit 7: Task 4
Name ________________________________
a. According to the College Board, Georgia seniors graduating in 2008 had a mean
Math SAT score of 493 with a standard deviation of 108. Assuming the distribution of
these scores is normal, find the probability of a member of the 2008 graduating class in
Georgia scoring between 500 and 800 on the Math portion of the SAT Reasoning Test.
b. According the same College Board report, the population of American 2008 high
probability that of a randomly selected senior from this population scoring between
500 and 800 on the Math portion of the SAT Reasoning Test.
c. Compare the probabilities from parts a and b. Explain the differences in the two
probabilities.