Download Chapter 6 Section 1 on Normal Distributions

Chapter 6 Section 1 on Normal Distributions
Name: _______________________________ Date: ______________ Period: _____
A continuous probability distribution for a given random variable is called a
_________________________________. It is bell shaped with mean = median = mode
in center of distribution.
There are 2 inflection points: μ-σ and μ+σ
The LARGER the σ the __________ area there will be in the tails of the distribution.
This curve will be _______________.
The SMALLER the σ the __________ area there will be in the tails of the distribution.
This curve will appear more ____________.
Total area under a curve = 1. The area under the curve to the left of the point equals the
___________________.
The x axis is a _________________ ______________________.
Properties of a Normal Distribution
1. symmetrical and _____________________
2. defined by ____ and _____
3. area under curve = _____
4. __________ acts as horizontal asymptote
Properties of a STANDARD Normal Distribution
1. symmetrical and ___________________
2. μ = _________ and σ = ________
3. area under curve = ______
4. __________ acts as horizontal asymptote
To standardize a normal curve, convert each x-value to a standard score.
Z score
z=
x–μ
σ
Given a normal curve μ = 48 and σ = 5, convert to a standard normal curve and
indicate where a score of x =45 would be on each curve.
Properties of Normal Distributions: Determine if the following statements are true or
false. If a statement is false, state the reason why.
1. There are a limited number of normal distributions.
2. There is only one standard normal distribution.
3. The mean of a normal distribution is always 0.
4. The mean of the standard normal distribution is always 0.
5. The standard deviation of the standard normal distribution is always 0.
6. For any normal distribution, the mean, median, and mode are equal.
7. The line of symmetry for a normal distribution is x = μ.
8. The y-axis is a vertical asymptote for all normal distributions.
9. The inflection points for any normal distribution are one standard deviation on
either side of the mean.
10. Normal distributions are symmetric, but do not have to be bell shaped.
Recognizing Normal Distribution:
For the following distributions, determine if the distribution is likely to be normal. If not,
explain why.
11. The weight of 500 American men.
12. The age at death of 200 gorillas studied in the wild.
13. The grades for 100 students on a relatively easy exam.
14. The length of 300 babies born in a Las-Vegas hospital.
15. The frequency of the outcomes on a roulette wheel.
Drawing Normal Distribution:
Draw the standard normal curve. Calculate the standard score of the given x value.
Indicate where the z value would be.
16. μ = 65 σ = 20 x = 40
17. μ = 5 σ = -0.25 x = 4.8
18. μ = 15 σ = 2 x = 19
19. μ = 0.023 σ = 0.001 x = 0.02
Statistics Chapter 6 Section 2
Reading a Normal Curve Table
Many of the distributions in which statisticians are interested are normal. One important
application of a normal distribution is that the ____________ under any part of the curve
is equal to the __________________ of the random variable falling within that region.
Since this information is continuous, including the endpoint of our range does not change
the value of the probability.
Generally, the area under a normal curve is figured using integration. Since we do not
want to rely on calculus to figure the area, we will use a
_________________________________, which is also called a _______________.
This table gives values that represent distance from the ____________ or ______.
Remember that the mean _______________________________.
1. Find the area under the standard normal curve to the left of z = 1.37.
2. Find the area under the standard normal curve to the right of z = 1.37.
3. Find the area under the standard normal curve between z = -1.68 and z = 2.
4. Find the area under the standard normal curve to the left of z = -2.5 and to the
right of z = 3.
5. Find the area under the standard normal curve to the left of z = -1.23 and to the
right of z = 1.23.
6. P(z < - 2.67)
7. P(z < 1.45)
8. P(1.25 < z < 2.31)
9. P(z < -2.5 or z > 2.5)
10. P(z > - 1.37)
11. P(z < -4.01)
12. P(z < 3.98)
13. P(z > 2.17)
Practice Problems:
Find the area to the LEFT of the given z value.
1. z = 2.35
2. z = -1.25
Find the area to the RIGHT of the given z value.
3. z = 1.35
4. z = -2.51
Find the area between the given z values.
5. z = 0.35 and z = 1.85
6. z = -1.25 and z = 2.16
Find the area less than –z and greater than z for each value of z given.
7. z = 1.46
8. z = -2.11
Find the combined area to the left of z1 and the right of z2.
9. z1 = -2.31, z2 = 1.67
10. z1 = -1.75, z2 = 1. 89
Find the given probabilities using the Normal Curve Table.
11. P(z < -3.14)
12. P(z > 2.72)
13. P(-1.86 < z < 3.14)
14. P(0.78 < z < 2.64)
15.P(z < -2.39 or z > 2.39)
Chapter 6 Section 3
Application problems using the normal curve
Decide which of the four probabilities you are using.
1. less than some value
2. greater than some value
3. between two values
4. less than one value and greater than another value.
Percentage is written with 2 decimal places, so you will have to have a probability
with four decimal places ALWAYS!!
1. Body temperatures of adults are normally distributed with a mean of 98.6˚F and a
standard deviation of 0.73˚F. What is the probability of having a normal body
temperature less then 96.9˚F?
2. Body temperatures of adults are normally distributed with a mean of 98.6˚F and a
standard deviation of 0.73˚F. What is the probability of having a normal body
temperature great than 100˚F?
3. Body temperatures of adults are normally distributed with a mean of 98.6˚F and a
standard deviation of 0.73˚F. What is the probability of having a normal body
temperature between 98˚F and 99˚F?
4. Body temperatures of adults are normally distributed with a mean of 98.6˚F and a
standard deviation of 0.73˚F. What is the probability of having a normal body
temperature less than 97.6˚F or greater than 99.6˚F?
Practice Problems:
1. Deviation IQ scores, sometimes called Wechsler IG scores, have a mean of 100
and a standard deviation of 15.
a. What percentage of the general population has deviation IQs lower than 85?
b. What percentage of the general population has deviation IQs larger than 130?
c. What percentage of the general population has deviation IQs between 90 and 110?
d. What percentage of the general population has deviation IQs less than 90 or
greater than 110?
2. Replacement times for CD players are normally distributed with a mean of 7.1
years and a standard deviation of 1.4 years.
a. Find the probability that a randomly selected CD player will have a replacement
time of less than 8.0 years.
b. Find the probability that a randomly selected CD player will have a replacement
time of more than 9.0 years.
c. Find the probability that a randomly selected CD player will have a replacement
time between 5 and 8 years.
d. Find the probability that a randomly selected CD player will have a replacement
time of less than 5.1 or greater than 9.1 years.
3. In a recent year, the ACT scores for high school students with and A or B grade
point average were normally distributed, with a mean of 24.2 and a standard
deviation of 4.2. A student with an A or B average that took the ACT during this
time is selected.
a. Find the probability that the student’s ACT scores is less than 20.
b. Find the probability that the student’s ACT score is greater than 31.
c. Find the probability that the student’s ACT scores is between 25 and 32.
d. Find the probability that the student’s ACT scores is less than 23.2 or greater than
25.2.
4. The heights of American women ages 18 to 24 are normally distributed with a
mean of 65 inches and a standard deviation of 2.4 inches. An American woman
in this age bracket is chosen at random.
a. What is the probability that she is more than 68 inches tall?
b. What is the probability that she is less than 70 inches tall?
c. What is the probability that she is between 64 and 69 inches tall?
d. What is the probability that she is less than 59 or greater than 71 inches tall?
5. According to the data released by the Chamber of Commerce of a certain city, the
weekly wages of office workers are normally distributed with a mean of $700 and
a standard deviation of $50. Consider a worker chosen at random from this city.
a. What is the probability that the worker makes a weekly wage of less than $600?
b. What is the probability that the worker makes a weekly wage of more than $810?
c. What is the probability that the worker makes a weekly wage of between $620
and $770?
d. What is the probability that the worker makes a weekly wage of less than $620 or
greater than $780?
6. The serum cholesterol levels in milligrams/deciliter (mg/dL) in a certain
Mediterranean population are found to be normally distributed with a mean of 160
and a standard deviation of 50. Scientists of the National Heart, Lung and Blood
Institute consider this pattern ideal for a minimal risk of heart attacks.
a. Find the percentage of the population who have blood cholesterol levels less than
150 mg/dL.
b. Find the percentage of the population who have blood cholesterol levels that
exceed the ideal level by at least 10 mg/dL.
c. Find the percentage of the population who have blood cholesterol levels between
150 and 200 mg/dL.
d. Find the percentage of the population who have blood cholesterol levels less than
100 mg/dL or greater than 220 mg/dL.
7. Monthly telephone bills in one region are normally distributed with a mean of $72
and a standard deviation of $14.
a. What is the probability that a phone bill chosen at random is less than $75?
b. What is the probability that a phone bill chosen at random is at least $90?
c. What is the probability that a phone bill chosen at random is between $80 and
$100?
d. What is the probability that a phone bill chosen at random is less than $58 or
greater than $86?
8. The average lifetime for a car battery is 148 weeks with a standard deviation of 8
weeks.
a. If a company guarantees its battery for 3 years, what percentage of the batteries
sold would you expect to be returned before the end of the warranty period?
Assume a normal distribution.
b. Imagine you were the CEO of the battery company. Evaluate the warranty offer
and list any changes you would make as the CEO.
9. Ella refuses to tell you her ACT score. However, she does tell you that her score
is above the mean. Which of the following z scores is possible for her ACT
score?
a. -0.08
b. 1.43
c. 0
d. not enough information
10. The average wage of first year graduates from nursing school is $34,000. At
Olivia’s job interview, she found out that she would make at most the mean
starting salary for first year graduates. Which of the following z scores are
possible for her wage?
a. 0
b. -1.42
c. 0.78
D. not enough information
Chapter 6 Section 4
Finding Z values using the normal curve
Using the z table in reverse to find scores based on probabilities.
1. What z value has an area of 0.7357 to its left?
2. Find the value of z such that the are to the left of z is 0.200.
3. What z value has an area of 0.0096 to its right?
4. Find the value of z such that the are between –z and z is 0.90.
5. Find the value of z such that the area to the left of –z plus the area to the right of z
is 0.1616.
6. What z value represents the 90th percentile?
7. The body temperatures of adults are normally distributed with a mean of 98.6
degrees F and a standard deviation of 0.73 degrees. What temperature represents
the 90th percentile?
Practice Problems: Find z values using the normal curve table.
1. What z value has an are of 0.0038 to its left?
2. What z value has an area of 0.9803 to its right?
3. What z value has na are of 0.0212 to its left?
4. What z value has an area of 0.0838 to its right?
5. Find the z value such that the are between –z and z is 0.99.
6. Find the z value such that the area between –z and z is 0.80.
7. Find the value of z such that the area to the left of –z and the area to the right of z
is 0.5686.
8. Find the value of z such that the are to the left of –z and the area to the right of z
is 0.0286.
9. What z value represents the 90th percentile?
10. What z value represents the 30th percentile?
11. What z value represents the 75th percentile?
12. What z value represents the 3rd quartile?
13. What z value represents the 1st quartile?
14. The body temeperatures of adults are normally distributed with a mean of 98.6
degrees F and a standard deviation of 0.73 degrees F. What temperature
represents the 85th percentile?
15. Assume that the salaries of elementary school teachers in the US are normally
distributed with a mean of $32,000 and a standard deviation of $6000. What is
the cutoff salary for teachers in the top 10%?
16. The Verbal Reasonsing, Biological Reasoning, and Physical Sciences section of
the MCAT are scores on a 1 to 15 scale with a mean of 8.0 and a standard
deviation of 2.5. If the medical school you are applying to only takes students
who score in the top 5%, what is the lowest scores you could make and still be
considered for acceptance?
17. Scores on midterm exams are normally distributed with a mean of 73 and a
standard deviatino of 9 points. A grade of ‘A’ is then given to students who score
at the 90th percentile or above. What is the minimum grade that will receive an
‘A’?
18. The numbers of monthly cell phone minutes used by students at one university are
normally distributed with a mean of 110 minutes and a standard devition of 33
minutes. What number represents the 30th percentil for cell phone usage at this
university?
19. A local firehouse received on average 45 calls per week with a standard deviation
of 6 calls. Suppose that the firefighters are anticipating an unusually busy week.
How many calls should they prepare for if they anticipate that this week will be
busier that 85% of their weeks?