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Sampling Distributions for a Mean a sample mean parameters such as population means dont have a dist

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Sampling Distributions for a Mean ... The sample mean has a different distribution. It's called the sampling distribution of the sample mean (for n = 5) ... – PowerPoint PPT presentation

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Title: Sampling Distributions for a Mean a sample mean parameters such as population means dont have a dist


1
Sampling Distributions for a Meana sample
mean parameters such as population means dont
have a distribution.
2
Word Lengths Gettysburg Address
  • The mean length is ? 4.295.
  • The standard deviation is ? 2.123.
  • Not Normal. Right skewed. The standard deviation
    isnt helpful for finding probabilities.

3
Sampling Distribution n 5
  • The sample mean has a different distribution.
    Its called the sampling distribution of the
    sample mean (for n 5).
  • The mean of these sample means is 4.165 (pretty
    close).
  • The standard deviation of these sample means is
    0.927 (less variable).
  • It fills the number line more completely.
  • The shape is unknown, but appears closer to
    Normal.

4
Sampling Distribution n 5
  • The mean of these sample means is 4.302 (very
    close).

The standard deviation of these sample means is
0.930.
The shape is close to Normal (but not Normal
theres right skew).
5
Sampling Distribution
  • Given A quantitative population with
  • mean ?
  • standard deviation ?
  • A random sample from the population, where the
    population is at least 20 times larger than the
    sample. (Independent trials.)
  • Statistic The sample mean .
  • This statistic is an unbiased estimate of the
    parameter ?.

6
Sampling Distribution Results
  • The distribution of the sample mean has
  • gt mean (means
    unbiased)
  • gt standard deviation
  • gt shape closer to Normal (but not
    necessarily Normal)

The book calls this the standard error of the
(sample) mean.
7
Sampling Distribution n 5
  • Example
  • Sample means from samples of size n 5 have
  • gt mean
  • gt standard deviation
  • gt shape closer to Normal (but not Normal
    a bit right skewed)

8
Sampling Distribution n 5
  • The mean of these sample means is 4.302 (very
    close to 4.295).

The standard deviation of these sample means is
0.930 (very close to 0.949).
The shape is close to Normal (but not Normal
theres right skew).
9
Sampling Distribution n 10
  • Example
  • Sample means from sample of size n 10 have
  • gt mean
  • gt standard deviation
  • gt shape closer to Normal

10
Sampling Distribution n 10
  • The mean of these sample means is 4.305 (very
    close).

The standard deviation of these sample means is
0.658.
The shape is very close to Normal (just a little
right skew not enough to fuss over).
11
Sampling Distribution
  • Example
  • Sample means from sample of size n 10 have
  • gt mean
  • gt standard deviation
  • gt shape closer to Normal
  • very close enough so that a Normal could be
    used for probabilities

12
Distribution of the Sample Mean
  • Given
  • A variable with population that is not Normally
    distributed with mean ? and standard deviation ?.
  • A random sample of size n.
  • Result
  • The sample mean has approximate Normal
    distribution with

Assume the population size is at least 20 times n.
13
Example
  • Rolls of paper leave a factory with weights that
    are Normal with mean ? 1493 lbs, and standard
    deviation ? 12 lbs.

14
Finding probabilities
  • What is the probability a roll weighs over 1500
    lbs?
  • ANS 0.2798
  • (about 28 of rolls exceed 1500 lbs)

15
New Question
  • A truck transports 8 rolls at a time. The legal
    weight limit for the truck is 12,000 lbs. What is
    the probability 8 rolls have total weight
    exceeding this limit?
  • Since 12000/8 1500, the question could also be
    phrased
  • What is the probability 8 rolls have (sample)
    mean weight exceeding 1500?
  • The bad news The answer is not 0.2798.
  • The good news Its not that tough.

16
Distribution of the Sample Mean
  • Given
  • A variable with population that is Normally
    distributed with mean ? and standard deviation ?.
  • A random sample of size n. (N/n ? 20)
  • Result
  • The sample mean has Normal distribution

Called the standard error of the sample mean.
17
Example - continued
  • Rolls (single rolls) of paper leave a factory
    with weights that are Normal with mean ? 1493
    lbs, and standard deviation ? 12 lbs.
  • If n 8 rolls are randomly selected, what is the
    probability their sample mean weight exceeds
    1500?
  • The distribution is Normal.

18
Finding probabilities
  • Find the probability the sample mean is over 1500
    lbs.
  • Here were using the same mean, but a standard
    deviation reduced to 4.243.
  • ANS 0.0495

19
Interpreting the Result
The probability the sample mean for 8 rolls
exceeds 1500 lbs is 0.0495. For 4.95 of all
possible samples of 8 rolls, the sample mean
exceeds 1500 lbs. Equivalent There is a 0.0495
probability that the total weight will exceed
8?1500 12,000 lbs. Again we are working with a
sampling distribution for a statistic. The
statistic here is the sample mean. Were working
towards using the sample mean as an estimate of
the population mean.
20
The Picture
Sample mean weights for samples of 8 rolls.
Weights of single rolls.
21
Example
  • Waiting times between customer arrivals at a
    service center have Exponential distribution with
    mean ? 2 minutes.
  • So ? 2 minutes. (Formula sheet.)
  • What can we say about the sample mean waiting
    time for n customers?
  • As n gets larger, the distribution gets closer to
    Normal.

22
The Picture
Single values
Sample mean n 64
Sample mean n 16
Sample mean n 4
23
The 20 Times Rule
Not close enough when n lt 20 times population
size. Too large. (There is an adjustment.)
24
Distribution of the Sample Mean
  • Given
  • A variable with population that is not Normally
    distributed with mean ? and standard deviation ?.
  • A random sample of size n.
  • Result
  • The sample mean has approximate Normal
    distribution with

Assume the population size is at least 20 times n.
25
Distribution of the Sample Mean
  • Given
  • A variable with population that is not Normally
    distributed with mean ? and standard deviation ?.
  • A random sample of size n.
  • Result
  • The sample mean has generally unknown
    distribution with

26
Distribution of the Sample Mean
Central Limit Theorem (CLT)
  • Given
  • A variable with population that is not Normally
    distributed with mean ? and standard deviation ?.
  • A random sample of size n, where n is
    sufficiently large.
  • Result
  • The sample mean has approximate Normal
    distribution with

27
What is Sufficiently Large?
  • Your book says generally n at least 30.
  • If the population is fairly symmetric without
    outliers, considerably less than 30 will do the
    trick.
  • If the population is highly skewed, or not
    unimodal, considerably more than 30 may be
    required.
  • If the population is Normal then sample size is
    not a concern The sample mean is Normal.
  • You may use the 30 rule if you recognize that
    its not that black and white, and that for
    Normal populations, n 1 is sufficiently large.

28
Example
  • The Census Bureau reports the average age at
    death for female Americans is 79.7 years, with
    standard deviation 14.5 years.
  • ? 79.7 years ? 14.5 years
  • I looked at a few recent obituaries in the Oswego
    Daily News (online)
  • 79 70 48 99 85 71 45

29
Example
  • Consider randomly samples of n 7 U.S. women
    What is the distribution of the sample mean?

30
Example
  • ? 79.7 years ? 14.5 years
  • The distribution of the sample mean has
  • Our sample has
  • How does this fit in?
  • Z (71.00 79.7) /5.48 -1.59
  • This suggests 71.00 is somewhat (but not very)
    unusually low.

31
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • Within 1 s.d.

32
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • If Normal
  • Within 1 s.d. (65, 95)

33
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • If Normal
  • Within 1 s.d. (65, 95) ? 68

34
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • If Normal
  • Within 1 s.d. (65, 95) ? 68
  • Within 2 s.d.s (50, 110) ? 95

35
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • If Normal
  • Within 1 s.d. (65, 95) ? 68
  • Within 2 s.d.s (50, 110) ? 95
  • Above 110

36
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • If Normal
  • Within 1 s.d. (65, 95) ? 68
  • Within 2 s.d.s (50, 110) ? 95
  • Above 110 ? 2.5
  • 1 in 40 ???
  • No way! The distribution is not Normal.

37
Example
  • The Normal shouldnt be used here (why not?)
  • Distribution of longevity ? ? 80 ? ? 15
  • If Normal
  • Within 1 s.d. (65, 95) ? 68
  • Within 2 s.d.s (50, 110) ? 95
  • Abover 110 ? 2.5
  • 1 in 40 ???
  • The distribution is not Normal.

38
Example
  • The Normal shouldnt be used here (why not?)

39
Example
  • The Normal shouldnt be used here (why not?)
  • The distribution of age at death is not Normal.
    It is quite left skewed.
  • The sample size is not sufficiently large. (At
    least 30 by your book, although for this
    situation your instructor would probably buy into
    as low as 20.)
  • ?
  • The Central Limit Theorem cant be applied.
  • The sample mean doesnt have approximate Normal
    distribution

40
Example
  • I looked at 41 more recent obituaries (total of
    48)
  • 79 70 48 99 85 71 45
  • more ?? data
  • 87 75 90 95 51 99 69
  • 71 49 93 80 89 77 72
  • 101 69 92 92 86 78 92
  • 89 91 81 74 68 89 92
  • 64 71 50 81 88 42 91
  • 44 51 85 81 92 93

41
Example
42
Example
  • What is the distribution of the sample mean of
    samples of size n 48?
  • Even though age at death is left skewed, with n
    48 (large enough) the Central Limit Theorem
    applies, and the sample mean has approximate
    Normal distribution.

43
Example
  • Normal
    My data
  • Find the probability that a random sample of 48
    U.S. womens deaths gives a sample mean 77.52 or
    less.
  • Z (77.52 79.7) / 2.09 -2.18 / 2.09 -1.04
  • Probability 0.1492
  • About 15 of all samples of 48 deaths give a
    sample mean 77.52 or less.

44
Example
  • The Census Bureau reports the average age at
    death for Americans is 79.7 years. My data on
    deaths for this region gave a mean of 77.52.
  • What most reasonably accounts for this difference
    of over 2 years?
  • CHANCE
  • Theres a 15 probability of this low or lower of
    a result for a sample drawn randomly from such a
    population. Theres a 30 probability of a result
    as far or farther from 79.7 (in either direction).

45
Example
  • The Census Bureau reports the average age at
    death for Americans is 79.7 years. My data on
    deaths for this region gave a mean of 77.52.
  • What most reasonably accounts for this difference
    of over 2 years?
  • H0 ?Oswego 79.7 H1 ?Oswego ? 79.7
  • Theres a 15 probability of this low or lower of
    a result for a sample drawn randomly from such a
    population. Theres a 30 probability of a result
    as far or farther from 79.7 (in either direction).

P-value!
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