SAMPLING DISTRIBUTIONS

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SAMPLING DISTRIBUTIONS CONFIDENCE INTERVAL

• CHAPTER 2
• BCT 2053

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CONTENT

• 2.1 Sampling Distribution
• 2.2 Estimate, Estimation and Estimator
• 2.3 Confidence Interval for the mean µ
• 2.4 Confidence Interval for the Difference
• between Two mean
• 2.5 Confidence Interval for the Proportion
• 2.6 Confidence Interval for the Difference
• between Two Proportions
• 2.7 Confidence Interval for Variances and
• Standard Deviations
• 2.8 Confidence Interval for Two Variances
• and Standard Deviations

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2.1 Sampling Distributions
OBJECTIVES

• After completing this chapter, you should be able
  to
• Identify the sampling distribution for sample
  mean, difference between two sample means, sample
  proportion and difference between two sample
  proportions.

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Sampling Distribution

• A sampling distribution is the probability
  distribution, under repeated sampling of the
  population, of a given statistic (a numerical
  quantity calculated from the data values in a
  sample).
• The formula for the sampling distribution depends
  on the distribution of the population, the
  statistic being considered, and the sample size
  used. A more precise formulation would speak of
  the distribution of the statistic for all
  possible samples of a given size, not just "under
  repeated sampling".
• In other word, the sampling distribution of a
  statistic S for samples of size n is defined as
  follows
• The experiment consists of choosing a sample of
  size n from the population and measuring the
  statistic S. The sampling distribution is the
  resulting probability distribution.

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Example

• Imagine that our population consists of only
  three numbers the number 2, the number 3 and the
  number 4. Our plan is to draw an infinite number
  of random samples of size n 2 and form a
  sampling distribution of the sample means.

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Sampling Distribution for Mean

• The mean of the sampling distribution of means is
  equal to the population mean.
• The standard deviation of the sampling
  distribution of means is
• for infinite population
• for finite population
• If the population is normally distributed, the
  sampling distribution is normal regardless of
  sample size.
• By using the Central Limit Theorem,
• If the population distribution is not
  necessarily normal, and has mean µ and standard
  deviation s , then, for sufficiently large n, the
  sampling distribution of is approximately
  normal, with mean       and
  standard deviation        

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Sampling Distribution for Different Mean
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Sampling Distribution for Proportions

• p - Proportion, Probability and
  Percent for population
• - sample proportion of x successes
  in a sample of
• size n
• - sample proportion of failures
  in a sample of size n
• x is the binomial random variable created by
  counting the number of successes picked by
  drawing n times from the population.
• The shape of the binomial distribution looks
  fairly Normal as long as n is large and/or p is
  not too extreme (not close to 0 or 1).

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Sampling Distribution for Proportions

• The sampling distribution for proportions is a
  distribution of the proportions of all possible n
  samples that could be taken in a given
  situation. 
• That is, the sample proportion (percent of
  successes in a sample), is approximately Normally
  distributed with
• mean p, and
• standard deviation

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Sampling Distribution for Different Proportions
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2.2 Estimate, Estimation, Estimator
OBJECTIVES

• After completing this chapter, you should be able
  to
• Define and understand the general formula of
  interval estimate (confidence interval) for a
  parameter.

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Estimator

• Probability function are actually families of
  models in the sense that each include one or more
  parameter.
• Example Poisson, Binomial, Normal
• Any function of a random sample whose objective
  is to approximate a parameter is called a
  statistic or an estimator
• is the estimator for

statistic
parameter
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Properties of Good Estimator

• Unbiased
• Efficient
• Sufficient
• Consistent

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Estimations Estimate

• Estimation Is the entire process of using an
  estimator to produce an estimate of the parameter
• 2 types of estimation
• Point Estimate
• A single number used to estimate a population
  parameter
• Interval Estimate
• A spread of values used to estimate a population
  parameter
• The interval is usually written (a, b) where a
  and b are known as confidence limit
• a lower confidence limit
• b upper confidence limit

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Definitions

• Confidence Interval
• Range of numbers that have a high probability of
  containing the unknown parameter as an interior
  point.
• By looking at the width of a confidence interval,
  we can get a good sense of the estimator
  precision.
• Width b a
• Confidence Coefficient ( )
• The probability of correctly including the
  population parameter being estimated in the
  interval that is produced

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Definitions

• Level of Confidence
• The confidence coefficient expressed as a percent
  ,
• Example

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Definitions
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2.3 Confidence Interval for Mean
OBJECTIVES

• After completing this chapter, you should be able
  to
• Find the confidence interval for the mean.
• Find the confidence interval for the mean when s
  is known and unknown.

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4.3 Confidence Interval for Mean
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Confidence Interval for the Mean
The ( 1 a ) 100 confidence interval for µ
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t- Distributions
The number of values that are free to vary after
a sample statistic has been computed
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Rounding Rule

• When you are computing a confidence interval for
  a population mean by using raw data, round off to
  one more decimal place than the number of decimal
  places in the original data.
• When you are computing a confidence interval for
  a population mean by using a sample mean and
  standard deviation, round off to the same number
  of decimal places as given for the mean.

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Example 1

• The mass of vitamin E in a capsule manufactured
  by a certain drug company is normally distributed
  with standard deviation 0.042 mg.
• A random sample of 5 capsules was analyzed and
  the mean mass of vitamin E was found to be 5.12
  mg.
• Find the 95 confidence interval for the
  population mean mass of vitamin E per capsule.

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Example 2

• A plant produces steel sheets whose weights are
  known to be normally distributed with a standard
  deviation of 2.4 kg.
• A random sample of 36 sheets had a mean weight
  of 3.14 kg.
• Find the 99 confidence interval for the
  population mean weight.

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Example 3

• A random number of 100 pieces of wood are cut
  using a machine.
• The sample mean of length in cm is 1.06 cm and
  the standard deviation is 0.08 cm.
• Find the 95 confidence interval for mean length
  all the woods cut by the machine.
• What is the width of this confidence interval?

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Example 4

• The mean IQ score for 25 UMP students is 115
  with standard deviation 10.
• If the IQ score for all UMP students is normally
  distributed.
• Find the 95 confidence interval for the mean IQ
  score for all UMP students.

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Example 5

• The result X of a stress test is known to be
  normally distributed random variable with mean µ
  and standard deviation 1.3.
• It is required to have a 95 confidence interval
  for µ with total width less than 2.
• Find the least number of tests that should be
  carried out to achieve this.

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Example 6

• 8 UMP students are randomly chosen and the value
  of their CPA has been collected as below.
• 3.20 2.76 2.94 3.41
• 2.92 2.99 3.01 3.11
• Find the 95 confidence interval for the CPA
  mean for all UMP students.

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Example 7

• The heights of men in a particular district are
  distributed with mean µ cm and the standard
  deviation s cm. On the basis of the results
  obtained from a random sample of 100 men from the
  district, the 95 confidence interval for µ was
  calculated and found to be (177.22 cm , 179.18
  cm). Calculate the value of sample mean and
  standard deviation.

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Example 8

• A 90 confidence interval for a population mean
  based on 144 observations is computed to be (2.7,
  3.4).
• How many observations must be made so that a 90
  confidence interval will specify the mean to
  within 0.2?

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2.4 Confidence Interval for the
Difference Between 2 Mean
OBJECTIVES

• After completing this chapter, you should be able
  to
• Find the confidence interval for the difference
  between two means when ss are known.
• Find the confidence interval for the difference
  between two means when ss are unknown and equal.
• Find the confidence interval for the difference
  between two means when ss are unknown and not
  equal.

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4.4 Confidence Interval for the
Difference Between 2 Mean
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Example 1

• The mean of sleep time for 50 IPTS students are
  7 hours with standard deviation of 1 hour.
• The mean of sleep time for 60 IPTA students is 6
  hours with standard deviation of 0.7 hour.
• Find the 99 confidence interval for the
  different mean of sleep time between the IPTS and
  IPTA students.
• Assume the population variance are same
• Assume the population variance are different

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Example 2

• The mean of sleep time for 20 IPTS students are
  7 hours with standard deviation of 1 hour.
• The mean of sleep time for 15 IPTA students is 6
  hours with standard deviation of 0.7 hour.
• Find the 99 confidence interval for the
  different mean of sleep time between the IPTS and
  IPTA students.
• Assume the population variance are same
• Assume the population variance are different

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Example 3

• Find the 95 confidence interval for the
  different mean of childrens sleep time and adults
  sleep time if given that the variances for
  childrens sleep time is 0.81 hours while for
  adults is 0.25 hours.
• The mean sample sleep time for 30 childrens are
  10 hours while for 40 adults are 7 hours.

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Example 4

• Two groups of students are given a problem
  solving test, and the results are compared. The
  data are follows
• Mathematics Majors Computer Science
  majors
• Find the 98 confidence interval for the
  different mean of test marks between the two
  groups of students. Assume the variance
  population test marks are same for both groups.

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Example 5

• A medical researcher wishes to see whether the
  pulse rates of nonsmokers are lower than the
  pulse rates of smokers. Samples of 110 smokers
  and 120 nonsmokers are selected. The results are
  shown here.
• Smokers Nonsmokers
• Find the 90 confidence interval for the
  different mean between pulse rates of nonsmokers
  and the pulse rates of smokers. Assume that the
  variance pulse rates for both populations are not
  same.

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2.5 Confidence Interval
for the Proportion
OBJECTIVES

• After completing this chapter, you should be able
  to
• Find the confidence interval for a proportion

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The ( 1 a ) 100 confidence interval for
proportion p
where
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Example 1

• 23 from 100 families in a village are poor. Find
  the 99 confidence interval poorness rate for
  this village.
• A survey was undertaken of the use of the
  internet by residents in a large city and it was
  discovered that in a random sample of 150
  residents, 45 logged on to the internet at least
  once a day. Calculate an approximate 90
  confidence interval for p, the proportion of
  residents in the city that log on to the internet
  at least once a day.

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Example 2

• Given . What sample size is
  needed to obtain a 95 confidence interval for p
  with width .
• A researcher whishes to estimate, with 90
  confidence, the proportion of people who own a
  home computer. A previous study shows that 40 of
  those interviewed had a computer at home. The
  researcher whishes to be accurate within 5 of
  the true proportion. Find the minimum sample size
  necessary.

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2.6 Confidence Interval
Difference between Two
Proportions
OBJECTIVES

• After completing this chapter, you should be able
  to
• Find the confidence interval for the difference
  between two proportions.

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The ( 1 a ) 100 confidence interval for the
different proportions p1 p2
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Examples

• Given
  . Find the 95 confidence intervals
  for .
• In a sample of 200 surgeons, 15 thought the
  government should control health care. In a
  sample of 200 general practitioners, 21 felt the
  same way. Find 95 confidence interval for the
  difference of proportions for surgeons and
  practitioners.

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2.7 Confidence Interval for Variances and
Standard Deviations
OBJECTIVES

• After completing this chapter, you should be able
  to
• Find the confidence interval for a variance and a
  standard deviation.

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The ( 1 a ) 100 confidence interval for the
variance
The ( 1 a ) 100 confidence interval for the
standard deviation
Where
(Chi-square distribution)
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Example 1

• A random sample of 10 rulers produce by a
  machine gives a group of data below, in cm.
• 100.13, 100.07, 100.02, 99.99, 99.88, 100.14,
  100.03, 100.10, 99.92, 100.21
• Find the 95 confidence interval for the height
  variance and standard deviation of all the rulers
  produce by the machine.

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Example 2

• A factory has a machine thats designed to
  filled boxes with an average of 24 ounces of
  cereal, and the population standard deviation for
  this filling process is expected to be 0.1 ounce.
  
• Thus, if the machine is working properly, the
  population variance should be 0.01 squared ounce.
  
• To estimate the value of population variance, an
  employee selected a random sample of 15 boxes
  from a supply filled by the machine and found
  that the sample variance was 0.008 squared ounce.
  
• Whats the 95 confidence interval for the
  population variance and standard deviation?

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2.8 Confidence Interval for Two Variances
and Standard Deviations
OBJECTIVES

• After completing this chapter, you should be able
  to
• Find the confidence interval for the confidence
  interval for the variance and a standard
  deviation proportion.

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The ( 1 a ) 100 confidence interval
for the variance proportion
Where
(F distribution)
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Example 1

• The machined in example 1 is serviced. A random
  sample of 12 rulers produces by the machine after
  the serviced made give a group of data below.
• 100.03, 100.01, 100.02, 100.04,
• 99.90, 99.96, 100.04, 100.06,
• 100.08, 99.98, 100.11, 100.05
• Find the 95 confidence interval for variance
  proportion for all rulers produces by the machine
  before and after the service.

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Example 2

• Before service, a machine can packed 10 packets
  of sugar with variance weight 64 g² while after
  service the variance weight for 5 packets of
  sugar are 25 g². Find the 99 confidence interval
  for variance proportion for all sugar produces by
  the machine before and after the service.

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Conclusion

• An important aspect of inferential statistics is
  estimation
• Estimations of parameters of populations are
  accomplished by selecting a random sample from
  that population and choosing and computing a
  statistic that is the best estimator of the
  parameter
• Statisticians prefer to use the interval estimate
  rather than point estimate because they can be
  95,99 or else confidence that their estimate
  contains the true parameter and also determine
  the minimum sample size necessary.

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Thank You