Confidence Interval Estimation for a Population Mean

1
Confidence Interval Estimation for a Population
Mean

• Lecture 38
• Section 10.3
• Wed, Mar 30, 2005

2
Confidence Intervals

• To estimate ?, we will use confidence intervals,
  as we did when estimating p.
• The basic form (as well as the theory) is the
  same
• (pt. est.) ? (approp. no. of st. devs.)

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Confidence Intervals

• If?x has a normal distribution, then the
  confidence interval is
• or
• ?
• If (?x ??)/(s/?n) has a t distribution, then
  the confidence interval is

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When to Use Z

• If
• The population is normal (or nearly normal) and ?
  is known (for any sample size), or
• The population is not normal, but the sample size
  is at least 30,
• Then use Z.

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When to Use t

• If
• The population is normal (or nearly normal), and
• ? is not known, and
• The sample size is less than 30,
• Then use t.

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Table IV

• Consider again the t table (Table IV).
• The degrees of freedom include every value up to
  30, then jump to 40, 60, 120.
• If the actual degrees of freedom are
• Between 30 and 40, use 30.
• Between 40 and 60, use 40.
• Between 60 and 120, use 60.
• If they are beyond 120, use z.

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Example

• See Example 10.5, p. 591 Cereal Boxes.
• Use Z. Why?
• n 25.
• ?x 9.82.
• Assume that ? 0.29.
• Level of confidence 95, so z 1.96.

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Example

• The confidence interval is
• 9.82 ? (1.96)(0.29/?25)
• 9.82 ? 0.114
• (9.706, 9.934).

9
TI-83 Confidence Intervals

• When the standard normal distribution applies, do
  the following.
• Press STAT.
• Select TESTS.
• Select ZInterval.
• A window appears requesting information.

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TI-83 Confidence Intervals

• Select Data or Stats.
• Assume we selected Stats.
• Enter ?.
• Enter?x.
• Enter n.
• Enter the level of confidence.
• Select Calculate and press ENTER.

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TI-83 Confidence Intervals

• A window appears containing
• The title ZInterval.
• The confidence interval in interval notation.
• The sample mean.
• The sample size.

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Lets Do It!

• Lets do it! 10.6, p. 593 How Much Beverage?

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Example

• See Example 10.6, p. 594 Empty Seats Imply
  Dollars Lost.
• Should we use Z or t?

14
TI-83 Confidence Intervals

• To use t, do the following.
• Press STAT.
• Select TESTS.
• Select TInterval.
• A window appears requesting information.

15
TI-83 Confidence Intervals

• Select Data or Stats.
• Assume we selected Stats.
• Enter?x.
• Enter s.
• Enter n.
• Enter the level of confidence.
• Select Calculate and press ENTER.

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TI-83 Confidence Intervals

• A window appears containing
• The title TInterval.
• The confidence interval in interval notation.
• The sample mean.
• The sample standard deviation.
• The sample size.