Download Let`s Be Normal Learning Task Until this task, we have focused on

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 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.
In the last task, dot plots 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 boxiness
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
five (𝜇 = 5) and standard deviation of one (𝜎 = 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.
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.
b) In this set, each curve has the same standard deviation. Determine the standard deviation and then
each mean.
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.
3. The SAT Verbal Test in recent years follow approximately a normal distribution with a mean of 505 (𝜇 =
505) and a standard deviation of 110 (𝜎 = 110). Joey took this test made a score of 600. The scores on the
ACT English Test are approximately a normal distribution with a mean of 17 (𝜇 = 17) and a standard deviation
of 2.5 (𝜎 = 2.5). Sarah took this test and made a score of 18. Which student made the better score? How do
you know?
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:
X 
z  X 
z 

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.
4. 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.
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.
5. We can also use z-scores to find the percentage or probability of events above a given observation.
a) The average on the most recent test Ms. Cox gave her French students was 73 with a standard
deviation of 8.2. Assume the test scores were normally distributed. Determine the probability that a
student in Ms. Cox’s class scored a 90 or more on the test.
b) Women’s heights are approximately normally distributed with 𝜇 = 65.5 inches and 𝜎 = 2.5 inches.
Determine the probability of a randomly selected woman having a height of at least 64 inches.
6. We can also determine the probability between two values of a random variable.
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 school graduates
had a mean Math SAT score of 515 with 𝜎 = 116. What is the 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.