chapter 2 part 1 sampling distribution
DESCRIPTION
hhTRANSCRIPT
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STA408 Pn. Sanizah's Notes 3/17/2014
1
SAMPLING DISTRIBUTION
AND
ESTIMATION
SAMPLING DISTRIBUTION
Population
Sample
Definition
The probability distribution of is called its sampling distribution. It lists the various values that can assume and the probability of each value of .
SAMPLING DISTRIBUTION OF MEANS
a distribution obtained by using the means computed from random samples of a specific size taken from a population
SAMPLING ERROR
the difference between the sample measure and the corresponding population
XX
X
Mean of the sample means will be the same as the population mean,
Standard deviation of the sample means will be smaller than the standard deviation of the population
x
Suppose a professor gave an 8-point quiz to a small class of 4 students. The results of the quiz were 2,6,4 and 8. For the sake of discussion, assume that the four students constitute the population.
The mean of population is:
= 5
The standard deviation of the population is:
= 5 = 2.2361
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STA408 Pn. Sanizah's Notes 3/17/2014
2
Now, if all samples of size 2 are taken with replacement and the mean of each sample is found. The distribution is as shown:
Sample Mean Sample Mean Sample Mean Sample Mean
(2,2) 2 (4,2) 3 (6,2) 4 (8,2) 5
(2,4) 3 (4,4) 4 (6,4) 5 (8,4) 6
(2,6) 4 (4,6) 5 (6,6) 6 (8,6) 7
(2,8) 5 (4,8) 6 (6,8) 7 (8,8) 8
A frequency distribution of sample means is as follows:
The mean of the sample means, is 5.
The standard deviation of sample means,
is 1.5811
X
X
Mean, 2 3 4 5 6 7 8
Freq,f
1 2 3 4 3 2 1X
If X1,X2,,Xn is a random sample of size, ntaken from population with mean, , and variance, 2 , then
a) the sample mean is
b) the sample variance is
The standard deviation of the sampling distribution is
X
XE )(
nXV
X
22)(
nn
2 Also known as standard error of
the mean
As the sample size n increases without limit, the shape of the distribution of the sample means taken with replacement from a population with mean and standard deviation will approach a normal distribution.
NOTE: provided that sample size, n is large (n30)
nNX
,~
If the sample size is sufficiently large, the central limit theorem can be used to answer questions about sample means in the same manner that a normal distribution can be used to answer questions about individual values.
NEW FORMULA must be used for the Z values:
n
XZ
/
nNXNX
22 ,~ then ),,(~ If
RULE 1
RULE 2
WHEN THE POPULATION DISTRIBUTION IS
NORMAL, THE SAMPLING DISTRIBUTION OF
X IS ALSO NORMAL FOR ANY SAMPLE SIZE nRULE 3
X
nX
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STA408 Pn. Sanizah's Notes 3/17/2014
3
A.C. Neilson reported that children between the ages 2 and 5 watch an average of 25 hours of television per week. Assume the variable is normally distributed and the standard deviation is 3 hours. If 20 children between the ages 2 and 5 are randomly selected, find the probability that the mean of the number of hours they watch television will be greater than 26.3 hours.
The average age of the vehicle registered in Malaysia is 8 years or 96 month. Assume the standard deviation is 16 months. If a random sample of 36 vehicles is selected, find the probability that the mean of their age is between 90 and 100 months.
The average time spent by construction workers on weekends is 7.93 hours (over 2 days). Assume the distribution is approximately normal with a standard deviation of 0.8 hours.
a. Find the probability an individual who works that trade works fewer than 8 hours on the weekend.
b. If a sample of 40 construction workers is randomly selected, find the probability that the mean of the sample will be less than 8 hours.
The average number of pounds of meat that aperson consumes a year is 218.4 pounds. Assumethat the standard deviation is 25 pounds and thedistribution is approximately normal.
a) Find the probability that a person selected at random consumes less than 224 pounds per year
b) If a sample of 40 individuals is selected, find probability that the mean of the sample will be less than 224 pound per year.