Download Binomial Random Variables

ACTIVITY SET 9 Feb. 15 Lab Stat 200 Sections 4-6
Activity 9.1 Suppose the amount students at PSU spent on textbooks this semester is a normal random
variable with mean μ= $360 and standard deviation σ = $90.
a. Use the empirical rule for bell-shaped data to determine intervals that will contain about 68%, 95% and
99.7% of amounts spent on textbooks by PSU students. Show work for each interval.
b. Draw a sketch of the probability density function for textbook amounts. Clearly label the horizontal
axis with the mean and values that indicate roughly the range of the amounts.
c. With Minitab, find the cumulative probability for $300. Use Calc>Probability
Distributions>Normal, enter the mean and standard deviation, click on “Input Constant” and enter 300
in the adjacent box. [Note: The notation for the probability you’re finding is P(X ≤ $300).]
d. Write a sentence interpreting the value found in the previous part that would be understood by
somebody with no training in statistics.
e. What proportion of students spent more than $535? (Start like you did for part c, but then you’ll have
to do a “by hand” calculation to determine the final answer. [Note: The notation for the probability you’re
finding is P(X > $535).]
f. What is the probability that a randomly selected student will have spent between $300 and $535? Show
work. [Note: You’ll be able to utilize information from parts c and e.]
g. Calculate a z-score for $400.
h. Use Table A.1 of the text to determine the cumulative probability for the z-score you calculated in the
previous part. If you don’t have the book with you, use the link to the table within the Week 6 folder.
i. Write a sentence interpreting the value found in the previous part that would be understood by
somebody with no training in statistics.
j. Use Minitab to find the probability a randomly selected student spent less than $400. If you didn’t get
the same answer as you did in part h, figure out why. The answers should be the same.
k. What proportion of students spent more than $400 on textbooks? Show any work.
l. Find the probability an amount spent is less than or equal to $240. Solve the problem using Table A1.
Show calculation of the z-score as part of your solution. (You can verify your answer using Minitab if
you wish.)
m. With Minitab, find the 75th percentile of amounts spent on textbooks. Toward the top of the dialog
box click “Inverse Cumulative Probability” and in the Input Constant box enter 0.75 (decimal fraction
version of 75%).
n. Write a sentence interpreting the value found in the previous part that would be understood by
somebody with no training in statistics.
o. In Table A1 search for the cumulative probability most near 0.75. You’re looking “inside” the table
where the probabilities are. What is the z-score with this cumulative probability?
p. Refer to the previous part. Determine the amount spent on textbooks having the z-score found in the
previous part. Show work. [Note: Compare your answer to the answer found in part m. If the answers
differ by a lot, something’s not right.]
Activity 9.2 Several years ago a military man wrote Dear Abby concerned that his wife gave birth 304
days after she had last seen him. He thought this was suspicious because the mean pregnancy length is
276 days. Suppose pregnancy length is a normal random variable with mean μ = 276 days and standard
deviation σ = 12.5 days. Provide numerical information in the form of a probability that would assure this
man that a 304-day pregnancy length is somewhat unusual, but really isn’t all that rare. Note: A few
different approaches are possible here.
Activity 9.4 A key concept in statistics is that different samples may give possibly different results.
Imagine we are interested in the proportion of times an event happens. Let n = sample size (or number of
trials) and p = true population proportion of the event (probability of success).
The possible value of the sample proportion is approximately a normal random variable with
p(1  p)
mean = p and
standard deviation =
.
n
a. Suppose we sample n= 400 individuals from a large population in which p = 0.40 (40%) is assumed to
be the proportion having a particular genetic trait. Assuming p = 0.4 is correct, what are the values of the
mean and standard deviation for the proportion with this trait in the sample of n =400?
mean =
standard deviation =
(give st.dev. to 4 decimal places)
b. With Minitab’s normal probability calculator, find the probability that the proportion of the sample
with the genetic trait will be less than or equal to 0.31. [Be sure to find a “cumulative probability.”]
c. Suppose we collect the sample data and observe that 0.31 (or 31%) of the sample has the genetic trait.
Explain why this is good evidence that the “true” population proportion might really be less than 0.40
(40%). Hint: What does answer in the previous part tell us?
d. Find a margin of error for this situation using the formula
1
and then explain how this value helps
n
us know that a sample result of 31% with the trait is too low for us to believe that the true percent is 40%.
e. Still assuming that the p = 0.40 value is correct “true” population proportion with the trait, what is the
probability that the proportion of the sample with the trait will be greater than 0.43? Use Minitab, but
you’ll have to do some “by hand” work too.