Understandable Statistics Eighth Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Edited by: Jeff, Yann, Julie, and Olivia

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Understandable StatisticsEighth Edition By
Brase and BrasePrepared by Lynn
SmithGloucester County CollegeEdited by Jeff,
Yann, Julie, and Olivia

• Chapter 8 Estimation

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Section 8.2

• Estimating µ When s Is Unknown

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Focus Points

• Learn about degrees of freedom and Students t
  distribution.
• Find critical values using degrees of freedom and
  confidence level.
• Compute confidence interval for µ when s is
  unknown. What does this information tell you?

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Statistics Quote

• The death of one man is a tragedy. The death of
  millions is a statistic.  Joe Stalin

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What if it is impossible or impractical to use a
large sample?

• Apply the Students t distribution.

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Properties of a Students t Distribution

1. The distribution is symmetric about the mean 0.
2. The distribution depends on the degrees of
   freedom, d.f. (d.f. n 1 for µ confidence
   intervals).
3. The distribution is bell-shaped, but has thicker
   tails than the standard normal distribution.
4. As the degrees of freedom increase, the t
   distribution approaches the standard normal
   distribution.

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Students t Variable
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The shape of the t distribution depends only only
the sample size, n, if the basic variable x has a
normal distribution.

• When using the t distribution, we will assume
  that the x distribution is normal.

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Table 6 in Appendix II gives values of the
variable t corresponding to the number of degrees
of freedom (d.f.)
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Students t Distribution Critical Values (Table
Excerpt)
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(No Transcript)
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Degrees of Freedom

• d.f. n 1
• where n sample size

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The t Distribution has a Shape Similar to that of
the the Normal Distribution
A Normal distribution
A t distribution
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The Students t Distribution Approaches the
Normal Curve as the Degrees of Freedom Increase
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Find the critical value tc for a 95 confidence
interval if n 8.
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Convention for Using a Students t Distribution

• If the degrees of freedom d.f. you need are not
  in the table, use the closest d.f. in the table
  that is smaller. This procedure results in a
  critical value tc that is more conservative in
  the sense that it is larger. The resulting
  confidence interval will be longer and have a
  probability that is slightly higher than c.

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Confidence Interval for the Mean of Small Samples
(n lt 30) from Normal Populations

• c confidence level (0 lt c lt 1)
• tc critical value for confidence level c, and
  degrees of freedom n - 1

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The mean weight of eight fish caught in a local
lake is 15.7 ounces with a standard deviation of
2.3 ounces. Construct a 90 confidence
interval for the mean weight of the population of
fish in the lake.
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Mean 15.7 ounces Standard deviation 2.3
ounces

• n 8, so d.f. n 1 7
• For c 0.90, Table 6 in Appendix II gives t0.90
  1.895.

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Mean 15.7 ounces Standard deviation 2.3
ounces.

• E 1.54
• The 90 confidence interval is
• 15.7 - 1.54 lt ? lt 15.7 1.54
• 14.16 lt ? lt 17.24

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The 90 Confidence Interval14.16 lt ? lt 17.24

• We are 90 sure that the true mean weight of the
  fish in the lake is between 14.16 and 17.24
  ounces.

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Summary Confidence Intervals for the Mean

• Assume that you have a random sample of size n
  from an x distribution and that you have computed
  x-bar and s. A confidence interval for µ is
• where E is the margin of error
• How do you find E? It depends on how much you
  know about the x distribution.

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Situation I (most common)

• You don't know the population standard deviation
  s. In this situation you
• use the t distribution with margin of error
• with d.f. n 1
• Guidelines If n is less than 30, x should have a
  distribution that is mound-shaped and
  approximately symmetric. It's even better if the
  x distribution is normal. If n is 30 or more, the
  central limit theorem (Chapter 7) implies these
  restrictions can be relaxed.

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Situation II (almost never happens!)

• You actually know the population value of s. In
  addition, you know that x has a normal
  distribution. If you don't know that the x
  distribution is normal, then your sample size n
  must be 30 or larger. In this situation, you use
  the standard normal z distribution with margin of
  error

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Which distribution should you use for x?
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Calculator Instructions

• CONFIDENCE INTERVALS FOR A POPULATION MEAN
• The TI-83 Plus and TI-84 Plus fully support
  confidence intervals. To access the confidence
  interval choices, press Stat and select TESTS.
  The confidence interval choices are found in
  items 7 through B.

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Example (s is unknown)

• A random sample of 16 wolf dens showed the number
  of pups in each to be 5, 8, 7, 5, 3, 4, 3, 9, 5,
  8, 5, 6, 5, 6, 4, and 7.
• Find a 90 confidence interval for the population
  mean number of pups in such dens.

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Example (s is unknown)

• In this case we have raw data, so enter the data
  in list using the EDIT option of the Stat key.
  Since s is unknown, we use the t distribution.
  Under Tests from the STAT menu, select item
  8TInterval. Since we have raw data, select the
  DATA option for Input. The data is in list and
  occurs with frequency 1. Enter 0.90 for the
  C-Level.

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Example (s is unknown)

• Highlight Calculate and press Enter. The result
  is the interval from 4.84 pups to 6.41 pups.

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Section 8.2, Problem 11

• Diagnostic Tests Total Calcium Over the past
  several months, an adult patient has been treated
  for tetany (severe muscle spasms). This condition
  is associated with an average total calcium level
  below 6 mg/dl (Reference Manual of Laboratory
  and Diagnostic Tests, F. Fischbach). Recently,
  the patient's total calcium tests gave the
  following readings (in mg/dl).
• 9.3 8.8 10.1 8.9 9.4 9.8 10.0
• 9.9 11.2 12.1
• Use a calculator to verify that x-bar 9.95 and
  s 1.02.
• (b) Find a 99.9 confidence interval for the
  population mean of total calcium in this
  patient's blood.
• (c) Based on your results in part (b), do you
  think this patient still has a calcium
  deficiency? Explain.

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Solution
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Section 8.2, Problem 17

• Finance P/E Ratio The price of a share of stock
  divided by the company's estimated future
  earnings per share is called the PIE ratio. High
  PIE ratios usually indicate growth stocks or
  maybe stocks that are simply overpriced. Low P/E
  ratios indicate value stocks or bargain stocks.
  A random sample of 51 of the largest companies in
  the United States gave the following P/E ratios
  (Reference Forbes).
• 11 35 19 13 15 21 40
  18 60 72 9 20
• 29 53 16 26 21 14 21
  27 10 12 47 14
• 33 14 18 17 20 19 13
  25 23 27 5 16
• 8 49 44 20 27 8 19
  12 31 67 51 26
• 19 18 32
• (a) Use a calculator with mean and sample
  standard deviation keys to verify that x-bar
  25.2 and s 15.5.

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Section 8.2, Problem 17

• (b) Find a 90 confidence interval for the P/E
  population mean µ of all large U.S. companies.
• (c) Find a 99 confidence interval for the P/E
  population mean µ of all large U.S. companies.
• (d) Bank One (now merged with J. P. Morgan) had a
  P/E of 12, ATT Wireless had a P/E of 72, and
  Disney had a P/E of 24. Examine the confidence
  intervals in parts (b) and (c). How would you
  describe these stocks at this time?
• (e) In previous problems, we assumed the x
  distribution was normal or approximately normal.
  Do we need to make such an assumption in this
  problem? Why or why not? Hint See the central
  limit theorem in Section 7.2.

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Solution