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LAB ACTIVITY 8
Due Friday, Oct. 14 at 11:59pm
Activity 1 (Does not require Minitab): For the following situations, decide which represent
binomial random variables. If a situation is binomial, specify n and p. If it is not, specify which
of the four conditions were not met.
1. X = number of heads from flipping the same coin ten times, where the probability of a
head = 1/2.
2. X = number of sixes from rolling two dice five times each, where the probability of a six
for one die is 1/6 and the probability of a six for the other die (which is weighted to be
unfair) is 1/2.
3. X = number of cities in which it will rain tomorrow among five neighboring cities located
within 10 miles of each other.
4. X = the number of times we flip a coin before getting the first occurrence of tails.
Activity 2 (Requires Minitab): Using Minitab to explore binomial distributions.
1. We roll a fair die 30 times and let X = the number of times that a two is rolled. Then X is
a binomial random variable.
a. What is n?
b. What is p? Write it as a decimal.
2. We can use Minitab to visualize the distribution of X using a distribution plot, which is
just like a histogram except it uses probabilities from a random variable instead of
observations from a sample.
To get the distribution plot for the X in part 1: graph  probability distribution plot 
view single  under the distribution drop down menu select ‘Binomial’  enter the ‘n’
from 1.a for Number of trials and ‘p’ from 1.b for Event probability  ok.
a. How would you describe this distribution plot:
Symmetric
Clearly skewed with mean > median
Clearly skewed with mean < median
b. Use the formula for the mean and standard deviation of a binomial random
variable to calculate the expected value and the standard deviation of X in this
example.
3. Now let’s change the example a bit. What if our die is weighted, and the probability of
rolling a two is actually only 0.05? Assume we still roll the die 30 times, and X = number
of times we roll a 2.
a. What is the expected value of X now?
b. Create the distribution plot for this random variable with p=0.05. Which of the
things below do you notice?
i. This new distribution plot is more skewed.
ii. This new distribution plot is more symmetric.
c. You can hover your mouse over a bar on the distribution plot and learn about the
probability that X takes on that value. Use this technique to find P(X=1).
4. Consider a new binomial random variable with n=100 and p=0.6. Create a distribution
plot for this random variable. Which of these terms best describes this distribution?
a. Symmetric, but not bell-shaped
b. Symmetric and bell-shaped
c. Skewed, with mean > median
d. Skewed, with mean < median
Activity 3 (Requires Minitab): Use Minitab to find the probabilities below, where Z stands for
a standard normal random variable. You may want to experiment a bit with Minitab’s
capabilities for helping find these probabilities; begin with Graph  Probability Distribution
Plots and take a look at slide 29 from Lecture 14 if you get stuck (try doing this on your own
before asking for help). Note that you may need to change “Right Tail” to “Left Tail” or
“Middle” depending on the question. Also, any statistical program, many modern calculators,
and numerous online websites will provide standard normal distribution calculators.
a. P(Z < 2.39) =
b. P(Z < –1.47) =
c. P(Z > 1.87) =
d. P(Z < –2.33 or Z > 2.33) =
Activity 4 (Can use Minitab but it is not required): Think of the empirical rule. Using the
empirical rule alone, you should be able to select the correct answers for this activity. In lab, of
course, you can calculate these probabilities exactly. On a test, however, you will need to use
your knowledge of the empirical rule alone.
1. P(–1< Z < 1.5) =
a. .961
b. .628
c. .824
d. .775
2. P(.25 < Z < 2) =
a. .84
b. .38
c. .49
d. .16
3. P(Z > 2.2)
a. .014
b. .028
c. .16
d. 0.0
Comment for the rest of the lab:
Any probability for an arbitrary normal random variable, X, can be written in terms of the
standard normal random variable, Z. For example, consider X with mean 10 and standard
deviation 3. Say we want to phrase P(5<X<11) in terms of X. We calculate two Z-scores:
• Z-score #1: (11–10) / 3 = 1 / 3 = 0.33
• Z-score #2: (5–10) / 3 = –5 / 3 = –1.67
Thus, we can write P(5<X<11) = P(–1.67<Z<0.33).
Activity 5 (Requires Minitab): You will need Minitab (or some other standard normal
distribution calculator or table) for this activity. The Stanford-Binet IQ test is a common exam
used to measure a person’s “intelligence quotient” or IQ. The scores are approximately normally
distributed with a mean, μ, of 100 and standard deviation, σ, of 16.
1. For the following prompts, write each probability in terms of X (the IQ score), and Z
(the standard normal random variable), then calculate the actual probability. The
probabilities are that…
a. You score less than 88 on the IQ test.
b. You score higher than 112
2. The College Boards, which are administered each year to many thousands of high
school students, are scored so as to yield a mean of 500 and a standard deviation of
100. These scores are approximately normally distributed.
a. What percentage of the scores can be expected to lie between 450 and 600?
b. An exclusive club wishes to invite those scoring in the top 10% on the College Boards to
join. What score is required to be invited to join the club? (We refer to this value as the
90th percentile.)
Activity 6 (Does not require Minitab): From a survey the following table gives the number of
classes students reported missing during the semester (represented by X) and the probability for
each of these X values.
X = missed classes
0
1
2
3
4
5
P(X = x)
0.56
0.23
0.13
0.02
0.01
1. Is X = missed days a discrete variable or a continuous variable?
2. What must be the value of P(X = 3)? (Hint: what must the sum of all probabilities be?)
3. Determine E(X), the mean number of missed days and provide an interpretation of this value.
(Another symbol for E(X) is μ)
4. Find P(1<X<5).