Inference Confidence intervals for the mean

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Inference Confidence intervals for the mean
Population Mean - µ
Sample mean X
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Point estimate for µ
Example Unknown µ-mean SAT score of
students In a random sample of n75
students 515
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Limitations of point estimator

• How reliable is this estimate?
• What value do we expect to get in another sample?
• An estimate without and indication of its
  variability is of little value!!! We would like
  to know precisely how far tends to be from
  the parameter of interest µ.

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Interval estimate for µ

• Specify an interval in which you think µ lies.
• We want to say something such as
• We are 95 confident that µ is between 505 and
  515

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• According to the Central Limit Theorem
• is approximately normal for large n

Standard error of
(1-a) confidence interval
(1-a)
a/2
a/2
Z 1-a/2
Z a/2
µ
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Example

• Suppose a student measuring the boiling
  temperature of a certain liquid observes the
  readings (in degrees Celsius) 102.5, 101.7,
  103.1, 100.9, 100.5, and 102.2 on 6 different
  samples of the liquid. He calculates the sample
  mean to be 101.82. If he knows that the standard
  deviation for this procedure is 1.2 degrees, what
  is the confidence interval for the population
  mean at a 90 confidence level?
• 1-a0.9

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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 the
  population mean

95 confidence interval for
100.86,102.78
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http//bcs.whfreeman.com/ips4e/pages/bcs-main.asp?
vcategorys00010n99000i99010.01o

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Population mean 8, Population SD 5

• Sample 1
• 1,1,2,2,4,4,4,5,6,7,7,7,8,8,9,9,11,11,13,13,14,14,
  15,16,16
• Mean 8.32, SD4.74
• Sample 2 Mean6.76, SD 4.73
• Sample 3 Mean8.48, SD 5.27
• Sample 4
• -3,-3,-2,0,1,2,2,4,4,5,7,7,9,9,10,10,10,
• 11,11,12,12,14,14, 14, 19
• Mean 7.16, SD5.93

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Practice Confidence Intervals

• 1. A manufacturer of pharmaceutical products
  analyzes a specimen from each batch of a product
  to verify the concentration of the active
  ingredient. The chemical analysis is not
  perfectly precise. Repeated measurements on the
  same specimen give slightly different results.
  The results of repeated measurements follow a
  normal distribution quite closely. The mean µ of
  the population of all measurements is the true
  concentration in the specimen. The standard
  deviation of this distribution is known to be s
  0.0068 grams per liter. The laboratory analyzes
  each specimen three times and reports the mean
  result. Three analyses of one specimen give the
  following concentrations. 0.8403 0.8363
  0.8447. Give a 95 confidence interval for the
  true concentration.

0.8327,0.8481
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• 2. Suppose that we conduct a survey of 19
  millionaires to find out what percent of their
  income the average millionaire donates to
  charity.  It is known that the standard deviation
  of the percent they donate to charity is 5. In
  the sample we discover that the mean percent is
  15.  Find a 95 confidence interval for the mean
  percent.

0.128,0.173
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• 3. 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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• 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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• 4.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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• 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

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• 5. 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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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
(half the length of the CI)
For the blood potassium example m3.5-3.40.1
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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 0.2.
• Julies potassium level is measured three times
  and the mean result is . Give a 99 confidence
  interval for Julies mean blood potassium level.

99 confidence interval for µ
3.1,3.7
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Example

• (b) 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?)
• For the blood potassium example
  m____________

0.1
she needs 27 blood tests to achieve a 99 CI
3.3,3.5.
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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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Practice

• 1. 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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• 2. 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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• 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