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Chapter 7B
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4. A State Dept. of Education is writing a state-wide math test, and by law must decide how many points
will count as a "failing score." The test consists of 30 True/False questions and 60 multiple choice
questions with 4 answer options. The total score (TS) will be equal to the number of true/false
items correct plus half the number of multiple-choice items correct. A decision has been made to
make the failing score the score that a student would be expected to get if they randomly guessed
on all the questions.
a) If a student is randomly guessing, the 30 True/False questions can be regarded as a binomial chance
experiment. If we define the random variable T = score from T/F items, what are the mean and
standard deviation of T for a random student who is guessing?
b) If a student is randomly guessing, the 60 multiple choice questions can be regarded as a binomial
chance experiment with four choices (A, B, C, D). If we define the random variable M = score from
MC items, what are the mean and standard deviation of the M for a random student who is
guessing?
c)
What is the mean and standard deviation of the random variable TS?
3. When driving the nation's highways Anna is known as something of a lead foot. The number of miles
per hour over the speed limit varies, but has a mean of 7.5 mph and standard deviation of 2 mph.
Unfortunately, the state in which she goes to college adjusts the fines so that the amount of the fine
is given by the formula: F = 10(MPH )+ 112 , where F is the amount of the fine, and MPH is the
number of miles over the speed limit. Let random variable F be the amount of her fine if she is
randomly stopped for speeding.
a) What is the mean of F?
b) What is the standard deviation of F?
Chapter 7B
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4. In the somewhat less than towering ski slopes of northeastern Iowa new skiers come to learn to ski.
On Big Bunny Slope skiers will fail to make the turn at Big Bend. On Little Bunny Slope, skiers will
sometimes tumble at Little Hill. The ski instructors send the new skiers down the two slopes in
groups of 30, wait a few moments, and then send the Ski Patrol Ambulance down after them,
stopping at Big Bend and then Little Hill. The skiers descend the slopes far enough apart that they
don't run into each other, so their spills are all independent.
a) The probability that a random new skier in the group will need to be carried to the First Aid Station
after a spill at Big Bend is 0.22. If we define the random variable B = number of new skiers needing
to be driven to the First Aid Station from Big Bend, we can model this situation as a binomial chance
experiment. What is the mean and standard deviation of B?
b) The probability that a random new skier in the group will need to be carried to the First Aid Station
after a spill at Little Hill is 0.14. If we define the random variable L = number of new skiers needing
to be driven to the First Aid Station from Little Hill, what is the mean and standard deviation of L?
c) The total number of injuries requiring the Ambulance, T. What are the mean and standard deviation
of the random variable T?
1. Determine the following areas under the standard normal (z) curve.
a) The area under the z curve to the left of 2.53
b) The area under the z curve to the left of –1.33
c) The area under the z curve to the right of 0.76
d) The area under the z curve to the right of –1.47
Chapter 7B
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e) The area under the z curve between –1 and 3
f)
The area under the z curve between –2.6 and –1.2
g) What values of - z and + z separate the middle 95% of the standard normal distribution from the
extreme 5%?
3. A gasoline tank for a certain model car is designed to hold 12 gallons of gas. Suppose that the actual
capacity of the gas tank in cars of this type is well approximated by a normal distribution with mean
12.0 gallons and standard deviation 0.2 gallons. What is the probability that a randomly selected car
of this model will have a gas tank that holds at most 11.7 gallons?
Chapter 7B
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Suppose IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
Sketch the distribution:
1. What is the probability that a person has an IQ score greater than 120?
2. What is the probability that a person has an IQ score between 110 and 130?
3. What is the 90th percentile of IQ scores?
4. 2% of IQ scores are above __________.
Chapter 7B
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Is the distribution normal?
Use the normal distribution to approximate these binomial distribution: