Download Statistics 7 - Applications of Normal Distributions

Math 20-2
Statistics: Lesson #7
Solving Problems Involving the Normal Distribution
Objective: By the end of the lesson, you should be able to:
- Solve problems that involve determining a probability, given a data point or z-score.
- Solve problems that involve determining a data point, given a probability.
Recall: We have used the 68-95-99% Rule to estimate percentages of data that fall over, under,
or between whole number z-scores.
But what if the data value we’re interested in doesn’t have a whole number z-score?
e.g. 1) Find the percent of data that has a z-score less than -1.37 in a normal distribution.
Key Point: You can find the exact percentage of data below, above, or between two z-scores by
using the normalcdf function on the calculator:
- Access the DISTR menu by pressing 2nd VARS.
- Choose 2: normalcdf(
- Enter values as follows:
normalcdf(lowest z-score, highest z-score)
For “less than” questions, use _____________ for a lowest z-score.
For “greater than” questions, use _____________ for highest z-score.
e.g. 2) Find the percent chance that a data value has the following z-scores. (Round to the
nearest hundredth of a percent).
a) between –2 and –0.58
b) z  1.37
Generally, we want to use this process in a real-life normally distributed situation. In these
problems, you can enter the given values directly into the graphing calculator as follows:
normalcdf(lowest data value, highest data value, mean, standard deviation)
(Still use  100000 for “greater than” and “less than” problems.)
Math 20-2
Statistics: Lesson #7
e.g. 3) A manufacturer has found that the life of its tires is normally distributed with a mean of
55 000 km and a standard deviation of 8000 km.
a) Find the percentage of tires that last longer than 53 000 km, to the nearest hundredth.
b) If 45 000 tires are produced, how many will last between 40 000 km and 65 000 km?
e.g. 4) Speeds of cars on a particular stretch of highway are normally distributed with a mean of
112 km/h and a standard deviation of 7 km/h. The speed limit on this highway is 110
km/h. Police allow motorists a l0% leeway before issuing a ticket. If 250 cars pass by,
how many would you expect to be ticketed for speeding?
Math 20-2
Statistics: Lesson #7
You can also solve problems that require you to work back the other way: starting with a
percentage of the data, and finding the z-score or data value.
Key Point: To find the z-score or data value if you’re given a percent use the invNorm function
on the calculator:
- Access the DISTR menu by pressing 2nd VARS.
- Choose 3: invNorm(
- Enter values as follows:
invNorm(probability to the left) (enter as a decimal between 0 and 1)
* This gives you the z-score.
- If it is a contextual problem, enter:
invNorm(probability to the left,  ,  )
* This gives you the data value.
e.g. 5) Find the z-score, rounded to 2 decimal places for each of the following. Draw a sketch of
each first.
a) 42.47% of the data is less than this z-score.
b) The lowest z-score that is in the top 5% of the population.
e.g. 6) A manufacturer of personal music players has determined that the mean life of the
players is 32.4 months, with a standard deviation of 6.3 months. They want to determine
what length of warranty to offer if they don’t want to replace any more than 1.5% of the
players.
a) Solve this problem using the graphing calculator.
b) If 10 000 personal music players are sold, how many could the manufacturer expect
to be returned under warranty?
Math 20-2
Statistics: Lesson #7
c) The manufacturer wants to offer the option of purchasing an extended warranty.
What length of extended warranty should they offer if they want to replace no more
than 20% of the players?
e.g. 8) The annual salaries at a certain company are normally distributed with a mean of $61 500
and a standard deviation of $6800. What would your salary be if you were earning within
the top 15% of salaries at this company?
Assignment:
Applications of the Normal Distribution Worksheet
p. 292-294 #6-9, 11, 13, 15-17, 20 (try #14 for a challenge!)