Download Advanced Functions and Modeling Lesson 3: In this lesson, you will

Advanced Functions and Modeling
Lesson 3: In this lesson, you will learn the characteristics of a normal
distribution and apply the concepts in real-life applications.
Data can be "distributed" (spread out) in different ways.
It can be spread out more on the left (left skewed)…
It can be spread out more on the right (right skewed)…
Or it can be all jumbled up…
But there are many cases where the data tends to be around a central
value with no bias left or right. This is called a “normal distribution” or a
“bell curve.”
In this lesson, we will focus on a normal distribution.
In a normal distribution, the “Empirical Rule” is used to describe the areas
under the curve over intervals that are one, two, or three standard
deviations from the mean.
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THE EMPIRICAL RULE
Approximately 68% of the data values fall between ±1
standard deviation from the mean.
Approximately 95% of the data values fall between ±2
standard deviations from the mean.
Approximately 99.7% of the data values fall between ±3
standard deviations from the mean.
Applications
1. A restaurant determined that the calories of its meals are normally
distributed. The mean is 680 calories with a standard deviation of 63
calories.
a. Give a range of calories that includes 95% of the restaurant’s
meals.
b. The cook creates a meal with 617 calories. What percentage of
meals has more calories than this meal?
2. The heights of the 880 students at East High School are normally
distributed with a mean of 67 inches and a standard deviation of 2.5
inches.
a. Approximately how many students are more than 72 inches tall?
b. What percent of the students are between 59.5 and 69.5 inches
tall?
3. A machine used to fill water bottles dispenses slightly different
amounts into each bottle. Suppose the volume of water in 120 bottles
is normally distributed with a mean of 1.1 liters and a standard
deviation of 0.02 liter.
a. Approximately how many bottles of water are filled with less than
1.06 liters?
b. What percent of the bottles have between 1.08 and 1.14 liters?
4. The heights of 32 peaks in a mountain range are normally distributed
with a mean of 10,200 feet and a standard deviation of 295 feet.
a. Approximately how many peaks are more than 10,495 feet tall?
b. What percent of peaks are between 9610 and 10,790 feet tall?
5. As part of a noise pollution study, researchers measured the sound
level in decibels of a busy city street for 30 days. According to the
study, the average noise was 82 decibels with a standard deviation of
6 decibels. Assume that the data are normally distributed.
a. If a normal conversation was held at about 64 decibels, determine
the number of hours during the study that the noise level was this
low.
b. Determine the percent of the study during which the noise was
between 76 decibels and 88 decibels.
6. Dion commutes 290 miles each week for work. His car averages 29.6
miles per gallon with a standard deviation of 5.4 miles per gallon.
Assume that the data are normally distributed.
a. Approximate the number of miles that Dion’s car gets for a gas
mileage of 35 miles per gallon or better.
b. For what percentage of Dion’s commute does his car have a gas
mileage between 24.2 miles per gallon and 40.4 miles per gallon?
7. The average wait in line for the 16,000 daily passengers of a roller
coaster is 72 minutes with a standard deviation of 15 minutes.
Assume that the data are normally distributed.
a. Determine the number of passengers who wait less than 60
minutes to ride the roller coaster.
b. Determine the number of passengers who wait more than 90
minutes to ride the roller coaster.
8. In 1995, scores on the SAT were re-scaled to their original mean of
500 with an approximate standard deviation of 100. SAT scores in
the general population have a normal distribution.
a. Approximately what percentage of the 1995 scores fell between
400 and 600?
b. Approximately what percentage of the 1995 scores fell below 300?
9. In 2001, the national mean ACT Math score was 20.7, with an
approximate standard deviation of 6. ACT scores in the general
population have a normal distribution.
a. Approximately what percentage of the 2001 scores were higher
than 26.7.
b. Approximately what ACT Math score would one need to make in
2001 to be ranked among the top 2.5% of all who took the test?
10. Birth weight of a healthy newborn baby is 3,500 grams with a
standard deviation 600 grams.
a. What percentage of healthy babies will be labeled as "low birth
weight" (less than 2,500 grams)?
b. What birth weight will be exceeded 90% of the time?