Download Practice Problems #7

MAT 135, Practice Problems #7, Fall 2016
Name ________________________________________
Instructions: These homework problems are largely compiled from questions students have asked in
past classes during review sessions. Since I have selected and phrased these questions, they are likely to
be more similar to exam-style questions. You may do these problems for extra points, over and above
the 70% needed to move on in MyLabsPlus (scores you will still need to get to take the online quizzes).
These homeworks do not replace MLP assignments. You may complete these problems on a separate
page and staple to this coversheet except for those problems where the coversheet provides sufficient
space. I strongly encourage you to ask questions in class about any problems you do not know how to
do.
1. When doing a probability problem, what should you look for when determining whether you
use 1) counting rules, 2) binomial distribution, 3) normal distribution?
2. The amount of medication in a randomly selected pill has a mean of 200 mg with a standard
deviation of 12 mg. If the least amount of medication needed to be effective is 175 mg, what is
the probability that a randomly selected pill will contain an effective amount?
3. State three properties of any normal distribution. How does the standard normal distribution
differ from other normal distributions?
4. A runner has an average time for mile races of 4m10s (250 seconds), with a standard deviation
of 10 seconds. Use the Empirical Rule to determine the probability of running a 4-minute mile
(or less)?
5. If the record for men of the mile run is about 3m43s, what is the z-score of that record for the
runner above? How likely is the runner to break that record?
6. Calculate the z-scores for the probabilities
indicated in the graphs to the right.
7. On the blank graph to the right, graph the normal
distribution with mean of 75 and standard
deviation of 7. Label three standard deviations on
each side of the mean. Shade the portion of the
distribution that represents the
probability of a score greater
than 89.
8. What are some keywords to
look for when selecting
between a probability
(particular) distribution
function, or a cumulative
distribution function (for
instance, when using the binomial distribution)? How is the normal distribution different?
9. Sandy’s daughter Caroline got a 92 on a test with a mean of 84 and a standard deviation of 8
points. Her twin sister Lisa has a different teacher and got an 89 on a test with a mean of 75 and
a standard deviation of 10. Which sister got the higher score?
10. What is the z-score if the probability outside
the two scores (same value with ± signs) is a
total of 28%?
11. Molly obtained a 92 on an IQ test administered
by her school. If the mean of the test is 100
and the standard deviation is 17, what z-score
is equivalent to Molly’s score?
12. The average height of women playing
professional volleyball is 71 inches with a
standard deviation of 2.2 inches. What is the
probability a randomly selected player is taller than 6’5’’?
13. What is the probability associated with the shaded region on
the graph to the left?
14. The shaded region above shows the probability associated with a normal distribution with a
mean of 75 and a standard deviation of 10. The top value in the shaded region corresponds to a
measurement of 80. Find the value of the probability of scoring less than 80 in this distribution.
15. Write the calculator syntax for the Normalcdf function in your calculator.
16. A variable is normally distributed with mean of 46 and standard deviation of 6. Use the
Empirical Rule to estimate what proportion of the population falls between 46 and 58.
17. Using the same information as above, what is the z-score of an individual with a 62 in this
variable?
18. Under what conditions can we approximate a binomial random variable as though it was a
normal random variable?
19. A variable is normally distributed with a mean of 11 and a standard deviation of 2.4. What is the
𝑧-score associated with the value 6?
20. Use the Empirical Rule for the data above, the probability of falling between 6.2 and 13.4 in the
distribution.
21. The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately
normal with a mean of 1252 chips and with a standard deviation of 129 chips.
a. What is the probability that a randomly selected bag contains between 1100 and 1400
chocolate chips?
b. What is the probability that a randomly selected bag contains fewer than 1000 chips?
c. What proportion of bags contain more than 1175 chocolate chips?
d. What is the percentile rank of a bag that contains 1050 chocolate chips?
22. A variable is normally distributed with a mean of 41.2 and a standard deviation of 12.3. A
researcher wants to examine the bottom 10% of the distribution more closely. What is the
value at the top of the bottom 10%? (i.e. What is the boundary between the bottom 10% and
the top 90%?) Sketch the graph that describes the situation.
23. A test has a mean score of 81 and a standard deviation of 6.4. A student scores on 90 on the
test. What is the probability of receiving this score or higher? What percentile does the score
represent?
24. The probability of an event is 26%. If we consider only the event and its complement and
sample 50 people, is this sample sufficiently large to use the normal approximation to the
binomial?
25. A particular high school has a mean GPA of 2.5 with a standard deviation of 0.9. What GPA score
represents the top 15% of the class?
26. What is the standard score of someone with a GPA of 3.9? What about 2.2?
27. Find the standard score that marks the boundaries of the middle 50% of a distribution.
28. The SAT is distributed normally with a mean of 1498 and a standard deviation of 199. Use the
Empirical Rule to determine the bounds on the middle 68% of scores? What are the bounds on
the middle 95% of scores?
29. A series of normal probability plots are shown below. Determine which ones are normal and
which ones appear to be a poor fit to a normal graph.
30. A variable is normally distributed with a mean of 0.71 and a standard deviation of 0.12. Use the
empirical rule to find the probability of being greater than 0.35 and less than 0.83.
31. A variable is normally distributed with a mean of 0.41 and a standard deviation 0.05. Use the
empirical rule to determine the probability of being greater than 0.51.
32.
Find the standard score associated with the 77th
percentile.
33.
If the mean of the dataset graphed to the left is
1400 and the standard deviation is 200, what observation
value is associated with the 77th percentile?
34. Find the percentile associated with 𝑧 =
−1.88.
35. A variable is distributed normally with a
mean of 1775 with a standard deviation of
191. What is the standard score associated
with 2186.
36. What properties are associated with all normal distributions? How does the standard normal
distribution differ from general normal distributions? Give an example of notation associated
with the normal distribution and explain what they mean.
37. Two data sets are shown to the right. Draw a normal probability plot
for each set of data.
38. The US Army requires women’s heights to be between 58 inches and
80 inches. What percentage of women meet that requirement if the
mean of women’s height is 64 inches and the standard deviation is 3
inches? How many women in the US could be denied admission to
the Army on account of their height is 2% of the population of 350
million has considered applying?
39. A stats professor gives a test. The scores are normally distributed
with a mean 25 points and a standard deviation of 5 points. She plans to curve the
scores.
a. If she curves by adding 50 points, what is the new mean and standard
deviation? How many scores of each letter grade are there? Report your answer in % of the
class.
b. Is this result fair?
c. Suppose she instead curved by selecting the top 10% to be As, the next 20% to be Bs, the
middle 60% are Cs, the bottom 10% is an F, and the rest are Ds. Is this result more fair?
Why or why not?
40. When we approximate a binomial distribution with a normal distribution, why do we convert
𝑃(𝑋 = 3) to 𝑃(2.5 ≤ 𝑋 ≤ 3.5)?