1. The central limit theorem states that the distribution of sample means approaches a normal distribution as sample size increases, even if the population is not normally distributed.
2. For a sample size of 30 or more, the distribution of sample means can be approximated as a normal distribution, allowing probabilities to be found using the normal distribution.
3. The mean of the distribution of sample means is the same as the population mean, and its standard deviation is the population standard deviation divided by the square root of the sample size.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.4: The Central Limit Theorem
1. Illustrate point and interval estimations.
2. Distinguish between point and interval estimation.
Visit the website for more services it can offer:
https://cristinamontenegro92.wixsite.com/onevs
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.4: The Central Limit Theorem
1. Illustrate point and interval estimations.
2. Distinguish between point and interval estimation.
Visit the website for more services it can offer:
https://cristinamontenegro92.wixsite.com/onevs
Chapter 5 part1- The Sampling Distribution of a Sample Meannszakir
Mathematics, Statistics, Population Distribution vs. Sampling Distribution, The Mean and Standard Deviation of the Sample Mean, Sampling Distribution of a Sample Mean, Central Limit Theorem
Lecture 5 Sampling distribution of sample mean.pptxshakirRahman10
Objectives:
Distinguish between the distribution of population and distribution of its sample means
Explain the importance of central limit
theorem
Compute and interpret the standard error of the mean.
Sampling distribution of
sample mean:
A population is a collection or a set of measurements of interest to the researcher. For example a researcher may be interested in studying the income of households in Karachi. The measurement of interest is income of each household in Karachi and the population is a list of all households in Karachi and their incomes.
Any subset of the population is called a sample from the population. A sample of ‘n’ measurements selected from a population is said to be a random sample if every different sample of size ‘n’ from the population is equally likelyto be selected.
For the purpose of estimation of certain characteristics in the population we would like to select a random sample to be a good representative of the population.
The set of measurements in the population may be summarized by a descriptive characteristic, called a parameter. In the above example the average income of households would be the parameter.
The set of measurements in a sample may be summarized by a descriptive statistic, called a statistic . For example to estimate the average household income in Karachi, we take a random sample of the population in Karachi. The sample mean is a statistic and is an estimate of the population mean.
Because no one sample is exactly like the next , the sample mean will vary from sample to sample ,and hence is itself a random variable.
Random variables have distribution ,and since the sample mean is a random variable it must have a distribution.
If the sample mean has a normal distribution ,we can compute probabilities for specific events using the properties of the normal distribution.
Consider the population with population mean = μ
and standard deviation = σ.
Next, we take many samples of size n, calculate the mean for each one of them, and create a distribution of the sample means.
This distribution is called the Sampling Distribution of Means.
Technically, a sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population.
Standard error of the
mean:
The quantity σ is referred to as the standard deviation .it is a measure of spread in the population .
The quality σ/n is referred to as the standard error of the sample mean .It is a measure of spread in the distribution of mean
A very important result of statistics referring to the sampling distribution of the sample mean is the Central Limit Theorem .
Central Limit Theorem:
Consider a population with finite mean and standard deviation . If random samples of n measurements are repeatedly drawn from the population then, when n is large, the relative frequency histogram for the sample means ( calculated from repeated samples)
1. 6.5 THE CENTRAL LIMIT
THEOREM
Chapter 6:
Normal Curves and Sampling Distributions
2. Page
296
The Distribution, Given x is normal
Theorem 6.1 For a Normal Probability
Distribution
Let x be a random variable with a normal
distribution whose mean is μ and whose standard
deviation is σ. Let be the sample mean
corresponding to random samples of size n taken
from the x distribution. Then the following are true:
a) The distribution is a normal distribution
b) The mean of the distribution is μ
c) The standard deviation of the distribution is
It states: the “x bar” distribution will be normal provided the x distribution
is normal.
3. Page
297
The Distribution, Given x is normal
The distribution can be converted to the
standard normal z distribution using the following
formulas
n is the sample size
μ is the mean of the x distribution
σ is the standard deviation of the x distribution
4. Page
298
Standard Error
The standard error is the standard deviation
of a sampling distribution. For the
sampling distribution,
The expression standard error appears in
printouts from statistical software and in some
publications and refers to the standard deviation
of the sampling distribution being used.
5. Page
299
The Central Limit Theorem
Theorem 6.2 For ANY Probability Distribution
If x possesses any distribution with mean μ and
standard deviation σ, then the sample mean
based on a random sample of size n will have a
distribution that approaches the distribution of a
normal random variable with mean μ and a
standard deviation
as n increases without limit.
7. The Central Limit Theorem
Summary of Important Points
Foralmost all x distributions, if n is 30 or larger,
the distribution will be approximately normal.
Ina few applications you may need a sample size
larger than 30 to get reliable results.
The larger a sample sixe becomes, the closer the
distribution gets to normal.
The distribution can be converted to a standard
normal distribution using formulas provided.
8. Using the Central Limit Theorem to
Convert to the Standard Normal
Distribution
Where n is the sample size (n ≥ 30)
μ is the mean of the x distribution
σ is the standard deviation of the x distribution
9. Page
How to Find Probabilities 301
Regarding x
Given a probability distribution of x values where
n = sample size
μ = mean of the x distribution
σ = standard deviation of the x distribution
1. If the x distribution is normal, then the distribution is
normal.
2. If the x distribution is not normal, and n ≥30, then by
the central limit theorem, the distribution is
approximately normal.
3. Convert to z
4. Use the standard normal distribution to find
corresponding probabilities of events regarding
10. Page
297
Example 12 - Probability
Suppose a team of biologists has been
studying the Pinedale children’s fishing pond.
Let x represent the length of a single trout
taken at random from the pond. This group of
biologists has determined that x has a normal
distribution with mean = 10.2 inches and
standard deviation = 1.4 inches.
11. Example 12 - Probability
a)What is the probability that a single trout taken
at
random from the pond is between 8 and 12
inches long?
Solution:
The probability is = Normalcdf(-1.57,1.29)
about 0.8433 that a = .843267
single trout taken at ≈.8433
random is between
8 and 12 inches
long.
12. Example 12 - Probability
b) What is the probability that the mean length
of five trout taken at random is between 8 and
12 inches?
Solution:
= Normalcdf(-3.49,2.86)
= .99764
≈.9976
The probability is about 0.9976 that the mean length
based on a sample size of 5 is between 8 and 12
inches.
13. Example 12 - Probability
c) Looking at the results of parts (a) and (b), we
see that the probabilities (0.8433 and 0.9976)
are quite different. Why is this the case?
Solution:
According to Theorem 6.1, both x and have a normal
distribution, and both have the same mean of 10.2 inches.
The difference is in the standard deviations for x and . The
standard deviation of the x distribution is = 1.4.
14. Page
Example 13 – Central Limit 300
Theorem
A certain strain of bacteria occurs in all raw milk.
Let x be milliliter of milk. The health department
has found that if the milk is not contaminated, then
x has a distribution that is more or less mound-
shaped and symmetrical. The mean of the x
distribution is
= 2500, and the standard deviation is = 300.
In a large commercial dairy, the health inspector
takes 42 random samples of the milk produced
each day. At the end of the day, the bacteria count
in each of the 42 samples is averaged to obtain
the sample mean bacteria count .
15. Example 13 – Central Limit
Theorem
a) Assuming the milk is not contaminated, what
is the distribution of ?
Solution: n = 42, and the central limit theorem
applies.
Thus, the distribution will be approximately normal
with a mean and standard deviation.
16. Example 13 – Central Limit
Theorem
b) Assuming the milk is not contaminated, what is
the probability that the average bacteria count
for one day is between 2350 and 2650 bacteria
per milliliter?
Solution:
= Normalcdf (-3.24, 3.24
The probability is 0.9988 that
= .998804
is between 2350 and 2650 ≈ .9988
17. Page
302
Bias and Variability
Whenever we use a sample as an estimate of a
population parameter we must consider bias and
variability.
Bias is an inclination or prejudice for or against one
person or group, especially in a way considered to be
unfair.
A sample statistic is unbiased if the mean of its
sampling distribution equals the value of the
parameter being estimated.
The spread of the sampling distribution indicates the
variability of the statistic. The spread is affected by
the sampling method and the sample size. Statistics
from larger random samples have spreads that are
smaller.
18. Bias and Variability
From the Central Limit Theorem:
The sample mean is an unbiased estimator of
the mean μ when n ≥ 30.
The variability of decreases as the sample size
increases.