The law of large numbers The central limit theorem

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The law of large numbers The central limit
theorem
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A Statistic and a parameter
Sample

• A statistic anything that we calculate from a
  sample, e.g.
• -The average SAT score in a sample of 35
  students
• S- the standard deviation of salaries in
  sample of 50 managers
• P-the proportion of smokers in a class of 80
  students.
• A parameter a feature of the population, e.g.
• µ-the mean SAT score of the whole population of
  students
• s-the standard deviation of salaries in the
  whole population of managers
• p-the proportion of smokers in the whole
  population, and so on.
• In this topic we concentrate a specific statistic
  - .
• First, if the sample size increases the mean
  in the sample approaches the unknown population
  parameter µ.
• Means include proportions

Population
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The law of large numbers

• Draw observations at random from any
  population with finite mean µ. As the number of
  observations drawn increases, the mean of
  the observed values gets closer and closer to the
  mean µ of the population.

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A Statistic and a parameter
Sample

• A statistic anything that we calculate from a
  sample, e.g.
• -The average SAT score in a sample of 35
  students
• S- the standard deviation of salaries in
  sample of 50 managers
• P-the proportion of smokers in a class of 80
  students.
• A parameter a feature of the population, e.g.
• µ-the mean SAT score of the whole population of
  students
• s-the standard deviation of salaries in the
  whole population of managers
• p-the proportion of smokers in the whole
  population, and so on.
• In this topic we concentrate a specific statistic
  - . Computing in many samples gives us a
  list of estimates for the unknown population
  parameter µ. If we plot these values in a
  histogram we would see that they form a close to
  normally shaped distribution.
• Means include proportions.

Population
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The central limit theorem

• Take a large (30 or more) random sample of
  size n from any population with mean µ and
  standard deviation s. The sample mean, X is
  approximately normal with mean µ and standard
  deviation

Note if the original population is exactly
Normal, than
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normal uniform
right skewed
Distribution of X Sampling distribution of
X n2 n5 n30
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Example

• 1. The time it takes a student to get to the
  university by bus every morning varies from day
  to day. The average of these times is 30 minutes
  and the standard deviation of the times is 10
  minutes.
• For 36 days the student recorded the time it
  took him to get to the university.
• Let be the average amount of time in this
  sample.
• The distribution of ___________________
• The mean of ________________________
• The standard deviation of ________________

Approximately normal
µ30
s10/v361.67
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Practice the sampling distribution of

• 1. The level of nitrogen oxide (NOX) in the
  exhaust of a particular car model varies with
  mean 0.9 grams per mile (g/mi) and standard
  deviation 0.15 g/mi. A company has 125 cars of
  this model in its fleet.
• Let denote the mean NOX nitrogen level in a
  sample of 125 cars. The distribution of is
  close to Normal with
• µ __________ and
• s ______________
• (b) What is the probability that the mean level
  of a sample of 125 will exceed 1.0?
• (c) What is the probability that the NOX level of
  a particular car would exceed 1.0?
• The distribution of NOX is not specified cannot
  compute the probability

0.9
0.15/v1250.0134
P(Xbar1)P(Z(1-0.9)/.0134)P(Z7.46)0
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• 2. Household size X in the U.S has mean µ2.6,
  and standard deviation s1.4.
• a. Does this imply that the population
  distribution is normal?
• No. Moreover, it is more reasonable to assume
  that most households will have about 1 or 2 or 3
  people, and a few households will be usually
  large.
• b. Take a random sample of 10 households. Find
  the probability that the mean household size
  exceeds 2.7.
• Cant be done sample size 10 is too small to
  expect the central limit theorem to guarantee an
  approximately normal distribution of
• c. Take a random sample of size 100 households.
  Find the probability that the mean household size
  exceeds 2.7.
• Now is approximately normal (2.6,
  1.4/v100), and so

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• 3. X has mean of 20 and a standard deviation
  of 5. The shape of the distribution of X is not
  specified. For a random sample of size 30
• (a) What can you say about the sampling
  distribution of the sample mean ?
• Distribution of ___________________
  ___________
• Mean of ____________________________
  __
• Standard deviation of
  ________________________________
• Find the probability that will exceed 20.75.

Approximately normal
µ20
s5/v30.91
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• 4. XN(µ20, s5). For a random sample of size
  n6, determine the
• (a) Distribution of
• normal
• (b) Mean of
• 20
• (b) Standard deviation of
• 5/v62.04

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• 5. In a certain population, Math SAT scores
  are N(500,100)
• a) Pick 1 student at random. What is the
  probability that her score X is between 490 and
  510?

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• b) Pick 25 students at random. What is the
  probability that their sample mean score is
  between 490 and 510?
• X has a mean 500 and standard deviation 100,
  therefore

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• c) Pick 400 students at random. What is the
  probability that their sample mean score is
  between 490 and 510.
• X has a mean 500 and standard deviation 100,
  therefore

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• 6. A bottling company uses a filling machine
  to fill plastic bottles with a popular cola. The
  bottles are supposed to contain 300 milliliters
  (ml). In fact, the contents vary, according to a
  normal distribution with mean µ298 ml and
  standard deviation s3 ml.
• What is the probability that an individual bottle
  contains less than 295 ml?
• X-content of a bottle, XN(298, 32)
• (b) What is the probability that the mean
  contents of the bottles in a six-pack is less
  than 295 ml.
• Before you answer, do you expect this
  probability to be (i)same (ii)higher
  (iii)lower than the probability in (a)?

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• 8. A candy company manufactures a candy bar,
  whose weight is supposed to be 50 grams. In fact,
  the weight varies according to a normal
  distribution with mean 49 grams and standard
  deviation 2 grams. The candy bars are then packed
  in bags of four units.
• What is the probability that an individual candy
  bar weighs less than 46 grams?
• XN(µ49, s2)

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• b. What is the probability that the mean weight
  of the candy bars in a bag is less than 46 grams?

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• c. It is known that this candy bar is the
  favorite of 20 of American children at the ages
  5-10 years old. What is the probability that the
  proportion of children preferring this candy bar
  in a random sample of 1000 American children at
  the ages 5-10 is between 18 and 22?