Sampling Distributions by L3Yaq6O

VIEWS: 15 PAGES: 23

									 Sampling
Distributions
 Parameter
• A number that describes the population
• Symbols we will use for parameters
  include
  m - mean
  s – standard deviation
  p – proportion (p)
  a – y-intercept of LSRL
  b – slope of LSRL
 Statistic
• A number that that can be computed from
  sample data without making use of any
  unknown parameter
• Symbols we will use for statistics include
  x – mean
  s – standard deviation
  p – proportion
  a – y-intercept of LSRL
  b – slope of LSRL
Identify the boldface values as parameter or statistic.

  A carload lot of ball bearings has mean
  diameter 2.5003 cm. This is within the
  specifications for acceptance of the lot by
  the purchaser. By chance, an inspector
  chooses 100 bearings from the lot that
  have mean diameter 2.5009 cm. Because
  this is outside the specified limits, the lot
  is mistakenly rejected.
    Why do we take samples
   instead of taking a census?

• A census is not always accurate.
• Census are difficult or impossible
  to do.
• Census are very expensive to do.
A distribution is all the
values that a variable can be.
The sampling distribution
of a statistic is the
distribution of values taken
by the statistic in all
possible samples of the
same size from the same
population.
 Consider the population – the
length of fish (in inches) in my
pond - consisting of the values

2, 7, 10, 11, 14
What is the mean
mx =standard
 and
       8.8
sx = 4.0694
deviation of this
  population?
Let’s take samples of size 2
(n = 2) from this population:
How many samples of size 2 are
possible?  C = 10
               5   2


    mx = 8.8              Find all mean
                       What is the10 of
                       these samples and
                         and standard
    sx = 2.4919        deviation sample
                       record theof the
                             means.
                        sample means?
Repeat this procedure with sample
size n = 3
How many samples of size 3 are
possible?  C = 10
               5   3

    mx = 8.8       What of these
                  Find allis the mean
                      and standard
                    samples and
                     deviation of the
   sx =          record the sample
          1.66132 sample means?
                       means.
What do you notice?
• The mean of the sampling distribution
  EQUALS the mean of the population.
           mx = m
• As the sample size increases, the standard
  deviation of the sampling distribution
  decreases.
          as n      sx
A statistic used to estimate a
parameter is unbiased if the
mean of its sampling
distribution is equal to the
true value of the parameter
being estimated.
Activity – drawing samples
General Properties
                               If n is more than
                               10% of the
Rule 1:      mx = m            population size
                               (N) then Rule 2
                               becomes:

             s                  sXN n
Rule 2: sx =             sX    
             n                n   N 1
  This rule is approximately correct as long
  as no more than 5% (10%) of the
  population is included in the sample
  General Properties
Rule 3:
 When the population distribution is
 normal, the sampling distribution of x
 is also normal for any sample size n.
General Properties
Rule 4: Central Limit Theorem
 When n is sufficiently large, the
 sampling distribution of x is well
 approximated by a normal curve, even
 when the population distribution is not
 itself normal.
  CLT can safely be applied if n exceeds 30.
Remember the army helmets . . .
EX) The army reports that the distribution of
head circumference among soldiers is
approximately normal with mean 22.8 inches
and standard deviation of 1.1 inches.
a) What is the probability that a randomly
selected soldier’s head will have a
circumference that is greater than 23.5 inches?

        P(X > 23.5) = .2623
b) What is the probability that a random
sample of five soldiers will have an
average head circumference that is greater
than 23.5 inches?
      Do you expect the probability to
                or less than the answer
      be moreWhat normal curve are
               part (a)? Explain
            to you now working with?


       P(X > 23.5) = .0774
If n is large or the population
distribution is normal, then


      x  mx   x m x
   z         s
        sx        n

has approximately a standard
normal distribution.
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 the length has a normal
distribution with mean of 10.2 inches and
standard deviation of 1.4 inches. What is
the probability that a single trout taken at
random from the pond is between 8 and
12 inches long?       P(8 < X < 12) = .8427
What is the probability that the mean
length of five trout taken at random is
between 8 and 12 inches long?

     Do you expect the probability to
   P(8< x <12) = .9978
       be more or less than the answer
             to part (a)? Explain
What sample mean would be at the 95th
percentile? (Assume n = 5)

             x = 11.23 inches
A soft-drink bottler claims that, on average, cans
contain 12 oz of soda. Let x denote the actual
volume of soda in a randomly selected can.
Suppose that x is normally distributed with
s = .16 oz. Sixteen cans are selected with a
mean of 12.1 oz. What is the probability that the
average of 16 cans will exceed 12.1 oz?

     P(x >12.1) = .0062
Do you think the bottler’s claim is correct?
No, since it is not likely to happen by chance alone & the
sample did have this mean, I do not think the claim that
the average is 12 oz. is correct.
A hot dog manufacturer asserts that one of its
brands of hot dogs has a average fat content of 18
grams per hot dog with standard deviation of 1
gram. Consumers of this brand would probably
not be disturbed if the mean was less than 18
grams, but would be unhappy if it exceeded 18
grams. An independent testing organization is
asked to analyze a random sample of 36 hot dogs.
Suppose the resulting sample mean is 18.4 grams.
Does this result indicate that the manufacturer’s
claim is incorrect?
             Yes, not likely to happen by chance alone.
What if the sample mean was 18.2 grams, would
you think the claim was incorrect? No

								
To top