Download Chapter 7 Notes - Clinton Public Schools

Chapter 7.1 Sampling Distributions
Central Limit Theorem
Name: _____________________________________________ Date: _______________ Period: _______
Sampling Distribution of Sample Means Activity
Part 1:
Description of Population: ________________________________________________
Population mean μ: _________
Population standard deviation σ: ____________
Sample size n: __________
Dotplot of years
Shape of Distribution: _________________
Collecting the Data
# of trials: 10
mean: ______________
st dev: ______________
# of trials: 25
mean: _______________
st dev: _______________
GROUP INFO for sampling distribution
mean: ______________
st dev: ______________
mean: _______________
st dev: _______________
Dotplot of each – distribution of each
CLASS INFO for sampling distribution
mean: ______________
st dev: ______________
mean: _______________
st dev: _______________
Dotplot of each – distribution of each
Analyzing the data
1. How does the shape of the distribution change as the number of means in the
sampling distribution increases?
2. How does the mean of the sampling distribution change as the number of
means increases? How does x compare to μ?
3. How does sx change as the number of means increases? How does sx
𝜎
compare to 𝑛 ?
√
4. What have you learned about the normality of the distribution for the
sampling distribution of means?
***One of the MOST important and useful concepts in statistics is the Central Limit
Theorem. It forms the foundation for estimating population parameters and
hypothesis testing.
Central Limit Theorem
For any given population with mean μ and standard deviation σ, a sampling
distribution of the sample means, with sample sizes of AT LEAST 30, will have the
following characteristics.
1. The sampling distribution will approximate a normal distribution
REGARDLESS of the shape of the original distribution. The larger the sample
size the better the approximation.
2. The mean of the sampling distribution μx equals the mean of the population
μ.
3. The standard deviation of the sampling distribution σx equals the standard
deviation of the population σ divided by the square root of the sample size.
Examples:
1. If the mean of a given sampling distribution is 85, what is the mean of the
population with a sampling size of 100?
2. If the standard deviation of a given population is 9 and a sampling
distribution is created from the population with a sampling size of 100, what
is the standard deviation of the sampling distribution?
3. Some health reports claim that the average cold lasts 7 days If 120 samples
of size 100 are taken from across the US, what would you expect the average
of the sampling distribution to be?
4. The Federal Reserve Bank of New York conducted a study and claims that the
inflation rates of American households had a population standard deviation
of 0.2 percentage points in 1996. If a sampling distribution is created using
samples of size 78, what would be the standard deviation of the sampling
distribution?
Chapter 7 Section 2
Central Limit Theorem with Population Means
1. Draw a picture that describes the information in the question using
normal distribution.
2. Convert the values in the problems to standard scores using the sampling
distribution mean and standard deviation.
3. Use the normal curve table and the z values to find the are under the
curve.
Examples
1. The body temperatures of adults are normally distributed with a mean of
98.6 and a standard deviation of 0.73. What is the probability of a sample of
36 adults having an average normal body temperature less than 98.3?
2. The body temperatures of adults are normally distributed with a mean of
98.6 and a standard deviation of 0.73. What is the probability of a sample of
40 adults having an average normal body temperature greater than 99?
3. In 2006, prices of women’s athletic shoes were normally distributed with a
mean of $75.15 and a standard deviation of $17.89. What is the probability
that the average price of a sample of 30 pairs of women’s athletic shoes will
differ from the population mean by more than $3.00?
4. The walking gait of adult males is normally distributed with a mean of 2.4
feet and a standard deviation of 0.3 feet. A sample of 34 men’s strides is
taken.
a. Find the probability that an individual man’s stride is less than 2.1 feet?
b. Find the probability that the mean of the sample taken is less than 2.1
feet?
c. Find the probability that the mean of the sample taken is more than 2.1
feet?
d. Find the probability that the sample mean differs from the population
mean by more than 0.06 feet?
5. The distribution of the number of people in line at a grocery store check out
has a mean of 3 and a variance of 9. A sample of 50 grocery lines is taken.
a. Calculate the probability that the sample mean of the line length is more
than 4.
b. Calculate the probability that the sample mean of the line length is less
then 2.5.
c. Calculate the probability that the sample mean differs from the
population mean by less than 0.5.
6. Intelligence is often cited as begin normally distributed with a mean of 100.0
and a standard deviation of 15.0.
a. What is the probability of a random person on the street having an
intelligence level less than 95?
b. If a random sample of 50 people is taken, what is the probability that
their mean intelligence level will be less than 95?
c. If a random sample of 50 people is taken, what is the probability that
their mean intelligence level will differ from the mean intelligent by more
than 5?
7. A tea bag manufacturer needs to place 2 g of tea in each bag. If the
machinery places an average of 2.6 g of tea in each bag with a standard
deviation of 0.3 g, what is the probability that a randomly chosen bag will
have between 2.0 and 2.8 g of tea?
8. A medical journal lists the average fetal heart rate of 140 beats per minutes
with a standard deviation of 12 bpm. In a sample of 200, what is the
probability that a fetal heart rate differs from the mean by more than 25
bpm?
9. The average wait time in a drive-thru chain is 193.2 seconds with a standard
deviation of 29.5 seconds. What is the probability that in a random sample of
45 wait times, the mean is between 185.7 and 206.5 seconds?
Chapter 7 Section 3
Population Proportions and the Central Limit Theorem
A _______________________________________________ is the percentage of the population
that has a certain characteristic. The _____________________________________________ is
the percentage of the sample that has that certain characteristic.
We denote sample proportions with
decimal places.
which is read “p hat” and is typically 2
In order to use the Central Limit Theorem for sample proportions, we must
make sure that the following conditions have been met.
If so, then the sampling distribution of sample proportions can be assumed to be
normal, thus allowing us to use the normal distribution and z scores to calculate
probability for population parameters.
The formula for z scores for population parameters is (not in the calculator!!)
Examples:
1. In a certain conservative precinct, 79% of the voters are registered
Republicans. What is the probability that in a sample of 100 voters from this
precinct, more than 68 of the voters would vote for the Republican
Candidate?
2. In another precinct across town, the population is very different. In this
precinct 81% of the voters are registered Democrats. What is the probability
that, in a sample of 100 voters from this precinct, less than 80 of the voters
would vote for the Democratic Candidate?
3. It is estimated that 7% of all Americans have diabetes. Suppose that a sample
of 74 Americans is taken. What is the probability that the proportion of the
sample that is diabetic differs from the population proportion by less than
1%?
4. It is estimated that 7% of all Americans have diabetes. Suppose that a sample
of 74 Americans is taken. What is the probability that the proportion of the
sample that is diabetic differs from the population proportion by more than
2%?
5. A large car dealership claims that 47% of their customers are looking to buy
a sport utility vehicle (SUV). A sample of 61 customers is surveyed. What is
the probability that les than 40% are looking to buy an SUV?
6. The local nursery is waiting for its spring annuals to be delivered, and 20% of
the plants ordered are petunias. If the first truck contains 120 plants packed
at random, what is the probability that no more than 30 of the plants are
petunias?
7. At one private college, 34% of students are business majors. Suppose that
260 students are randomly selected from a list in the registrar’s office. What
is the probability that the proportion of business students in the sample
differs from the population proportion by less than 2%?
8. At a large grocery store, 72% of shoppers are women. In order to obtain
information about spending habits, 40 shoppers are randomly chosen for a
survey. What is the probability that the proportion of women in the sample
differs from the mean by more than 3%?
Chapter 7 Section 4
Approximating the Binomial Distribution Using the Normal Distribution
If the conditions np≥ 5 and n(1 – p) ≥ 5 are met for a given binomial distribution,
then a normal distribution can be used to approximate its probability
distribution with a given mean and a standard deviation.
μ = np
σ = √𝑛𝑝( 1 − 𝑝)
So when we need to calculate a probability for a large value of a binomial
random variable, we do not need to use the binomial formula (calculators cannot
calculate this for large values of n). Instead we use the normal distribution.
We are trying to use the normal distribution, which is ________________________ to
approximate the binomial distribution which is _____________________.
In order to do this a CONTINUITY CORRECTION must be used to convert the
whole number value of the discrete binomial random variable to an interval
range of the continuous normal random variable by using x + 0.5 and x – 0.5.
So…. If the value includes the number, it includes the interval. If it doesn’t
include the number, then it doesn’t include the interval.
Examples:
1. Use the continuity correction factor to describe the are under the normal
curve that approximates the probability that at least 2 people in a statistics
class of 50 cheated on the last test. Assume that the number of people who
cheated is binomial distribution with a mean of 5 and a standard deviation of
2.12.
2. Use the continuity correction factor to describe the are under the normal
curve that approximates the probability that less than 5 of the sitcoms
playing on TV tonight are reruns. Assume that the number of reruns on TV
tonight is a binomial distribution with a mean of 7 and a standard deviation
of 2.16.
3. Calculate the probability of more than 55 girls being born in 100 births.
Assume that the probability of a girl born in an individual birth is 50%.
4. After many hours of studying for your statistics test, you believe that you
have a 90% probability of answering any given question correctly. Your test
includes 50 true/false questions. What is the probability that you will miss
no more than 4 questions?
5. Many toothpaste commercials advertise that 3 out of 4 dentists recommend
their brand of toothpaste. What is the probability that out of a random
survey of 400 dentists, 300 will have recommended Brand X toothpaste?
Assume that the commercials are correct, and therefore, there is a 75%
chance that any given dentist will recommend Brand X.
6. What is the probability that more than 150 out of the 230 eighth graders at a
local middle school have been exposed to drugs? Assume that a previous
study at this school reported that the probability of an individual eighth
grade student having been exposes to drugs is 63%.
7. What is the probability that more than 100 out of 300 elections will contain
voter fraud? One report suggests that there is a 32% chance of an individual
elections containing voter fraud.
8. What is the probability that more than 20 out of a class of 347 high school
seniors will drive under the influence of alcohol on prom night. The local
chapter of MADD fears that the probability of a high school senior drinking
and driving on Prom night is 38%.
9. What is the probability that at least 67 out of 100 cars stopped at a roadblock
will not be given a ticket? Local authorities report that tickets usually are
given to 23% of cars stopped.
10. What is the probability that no more than 130 out of 2300 tax returns filed at
a local CPA’s office will be inaccurate? Previous records indicate only a 7%
probability that a given tax return from this office in inaccurate?