Confidence intervals for the mean - continued

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Confidence intervals for the mean - continued
Population Mean - µ
Sample mean X
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Reminder

• Point estimator for µ
• Limitations of point estimators
• Interval estimation for µ

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A (1-a) confidence interval for µ

• A (1-a) confidence interval for µ is

(1-a)
a/2
a/2
Z 1-a/2
Z a/2
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Example

• A test for the level of potassium in the blood
  is not perfectly precise. Moreover, the actual
  level of potassium in a persons blood varies
  slightly from day to day. Suppose that repeated
  measurements for the same person on different
  days vary normally with s0.2.

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• (a) Julies potassium level is measured three
  times and the mean result is .
  Give a 90 confidence interval for Julies mean
  blood potassium level.

90 confidence interval for
3.21,3.59
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• (b) A confidence interval of 95 level would be
• (i) wider than a confidence interval of 90
  level
• (ii) narrower than a confidence interval of 90
  level
• (c) Give a 95 confidence interval for Julies
  mean blood
• potassium level.

95 confidence interval for
3.174,3.626
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• (d) Julie wants a 99 confidence interval of
  3.3,3.5. What sample size should she take to
  achieve this (how many times should she measure
  her potassium blood level?)
• 3.3,3.53.4Z0.995(0.2/vn)
• 3.4-Z0.995(0.2/vn)3.3
• 3.4Z0.995(0.2/vn)3.5
• solve for n
• subtract the first equation from the second ?
  2Z0.995(0.2/vn)0. 2
• Z0.995(0.2/vn)0.1
• 2.575(0.2/vn)0.1
• (0.2/vn)0.0388
• vn5.15
• ? n26.5225
• she needs 27 blood tests to achieve a 99 CI
  3.3,3.5.

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Finding n for a specified confidence interval
Suppose we want a specific interval with a
confidence level 1-a. What sample size should be
taken to obtain this CI? Define m the distance
from the mean to the upper/lower limit of the CI
For the blood potassium example m3.5-3.40.1
(3.3-3.40.1)
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Example

• An agricultural researcher plants 25 plots with
  a new variety of corn. The average yield for
  these plots is 150 bushels per Acre.
  Assume that the yield per acre for the new
  variety of corn follows a normal distribution
  with unknown µ and standard deviation s10
  bushels per acre. A 90 confidence interval for µ
  is
• (a) 1502.00
• (b) 1503.29
• (c) 1503.92
• (d) 15032.9

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• An agricultural researcher plants 25 plots with
  a new variety of corn. The average yield for
  these plots is 150 bushels per Acre.
  Assume that the yield per acre for the new
  variety of corn follows a normal distribution
  with unknown µ and standard deviation s10
  bushels per acre. Which of the following will
  produce a narrower confidence interval than the
  90 confidence interval that you computed above?
• (a) Plant only 5 plots rather than 25
• (b) Plant 100 plots rather than 25
• (c) Compute a 99 confidence interval rather than
  a 90 confidence interval.
• (d) None of the above

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Example

• You measure the weight of a random sample of 25
  male runners. The sample mean is 60
  kilograms (kg). Suppose that the weights of male
  runners follow a normal distribution with unknown
  mean µ and standard deviation s5 kg. A 95
  confidence interval for µ is
• (a) 59.61,60.39
• (b) 59,61
• (c) 58.04,61.96
• (d) 50.02,69.8

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• You measure the weight of a random sample of 25
  male runners. The sample mean is 60
  kilograms (kg). Suppose that the weights of male
  runners follow a normal distribution with unknown
  mean µ and standard deviation s5 kg. Supposed I
  had measured the weights of a random sample of
  100 runners rather than 25 runners. Which of the
  following statements is true?
• (a) The lengths of the confidence interval would
  increase
• (b) The lengths of the confidence interval would
  decrease
• (c) The lengths of the confidence interval would
  stay the same
• (d) s would decrease

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Example

• Suppose we wanted a 90 confidence interval of
  length 4 for the average amount spent on books
  by freshmen in their first year at a major
  university. The amount spent has a normal
  distribution with s30.
• The number of observations required is closest
  to
• 25
• 30
• 609
• 865

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• mhalf of the CI length (marginal error)
• m
• a

4/22
0.1
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Example

• You plan to construct a confidence interval for
  the mean µ of a normal population with known
  standard deviation s. Which of the following will
  reduce the size of the confidence interval?
• use a lower level of confidence
• Increase the sample size
• Reduce s
• All the above

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Example

• A 95 confidence interval for the mean µ of a
  population is computed from a random sample and
  found to be 93. We may conclude that
• (a) There is a 95 probability that µ is between
  6 and 12
• (b) There is a 95 probability that the true mean
  is 9 and there is a 95 probability that the true
  mean is 3
• (c) If we took many additional random samples and
  from each computed a 95 confidence interval for
  µ, approximately 95 of these intervals would
  contain µ.
• (d) All of the above

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Example

• The heights of young American women, in inches,
  are normally distributed with mean µ and standard
  deviation s2.4. I select a simple random sample
  of four young American women and measure their
  heights. The four heights, in inches, are
• 63 69 62 66
• Based on these data, a 99 confidence interval
  for µ, in inches, is
• (a) 651.55
• (b) 652.35
• (c) 653.09
• (d) 654.07

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• Mean of four heights
• CI
• CI61.91,68.09

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• The heights of young American women, in inches,
  are normally distributed with mean µ and standard
  deviation s2.4. I select a simple random sample
  of four young American women and measure their
  heights. The four heights, in inches, are
• 63 69 62 66
• If I wanted the 99 confidence interval to be
  1 inch from the mean, I should select a simple
  random sample of size
• 2
• 7
• 16
• 39

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• mhalf of the CI length (marginal error)
• m
• a

1
0.01