Confidence Intervals

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Confidence Intervals

• Week 10
• Chapter 6.1, 6.2

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What is this unit all about?

• Have you ever estimated something and tossed in a
  give or take a few after it?
• Maybe you told a person a range in which you
  believe a certain value fell into.
• Have you ever see a survey or poll done, and at
  the end it says /- 5 points.
• These are all examples of where we are going in
  this section.

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Chapter 6.1 - disclaimer

• To make this unit as painless as possible, I will
  show the formula but will teach this unit with
  the use of the TI 83 graphing calculator
  whenever possible.
• It is not always possible to use the TI-83 for
  every problem.
• You can also follow along in Chp. 6.1 in the TEXT
  and use their examples in the book to learn how
  to do them by hand.

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What is a Confidence Interval?

• If I were to do a study or a survey, but could
  not survey the entire population, I would do it
  by sampling.
• The larger the sample, the closer the results
  will be to the actual population.
• A confidence interval is a point of estimate
  (mean of my sample) plus or minus the margin of
  error.

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What will we need to do these?

• Point of estimate mean of the random sample
  used to do the study.
• Confidence Level percentage of accuracy we need
  to have to do our study.
• Critical two-tailed Z value - (z-score) using
  table IV.
• Margin of Error a formula used involving the Z
  value and the sample size.

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Formula for Confidence Intervals

• This formula is to be used when the Mean and
  Standard Deviation are known

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Finding a Critical Z-value

• (Ex 1) Find the critical two-tailed z value for
  a 90 confidence level
• This means there is 5 on each tail of the
  curve, the area under the curve in the middle is
  90. Do Z (1-.05) Z .9500
• We will be finding the z score to the left of
  .9500 in table IV. It lands in-between
  .9495-.9505, thus it is /- 1.645
• (this is the 5 on each end)

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Finding a Critical Z-value
(Ex 2) Find the critical two-tailed z value for
a 95 confidence level This means there is
2.5 on each tail of the curve, the area under
the curve in the middle is 95. Do Z (1-.025) Z
.9750 We will be finding the z score to the left
of .9750 in table IV. It is /- 1.96 (this
is the 2.5 on each end)
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Finding a Critical Z-value
(Ex 3) Find the critical two-tailed z value for
a 99 confidence level This means there is
.005 on each tail of the curve, the area under
the curve in the middle is 99. Do Z (1-.005) Z
.9950 We will be finding the z score to the left
of .9950 in table IV. It is /- 2.575
(this is the .005 on each end)
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Finding a Critical Z-value
(Ex 4) Find the critical two-tailed z value for
a 85 confidence level
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Margin of Error

• The confidence interval is the sample mean, plus
  or minus the margin of error.

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Find the MoE

• Ex (5) After performing a survey from a sample
  of 50 mall customers, the results had a standard
  deviation of 12. Find the MoE for a 95
  confidence level.

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Special features of Confidence Intervals

• As the level of confidence () goes up, the
  margin of error also goes up!
• As you increase the sample size, the margin of
  error goes down.
• To reduce the margin of error, reduce the
  confidence level and/or increase the sample size.
• If you were able to include the ENTIRE
  population, the would not be a margin of error.
• The magic number is 30 samples to be considered
  an adequate sample size.

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Finding Confidence Intervals

• (Ex 6) After sampling 30 Statistics students at
  NCCC, Bob found a point estimate of an 81 on
  Test 3, with a standard deviation of 8.2. He
  wishes to construct a 90 confidence interval for
  this data.

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How did we get that?
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Using TI-83 to do this

• Click STAT
• go over to TESTS
• Click ZInterval
• Using the stats feature, input S.D., Mean, sample
  size, and confidence level.
• arrow down, and click enter on calculate.

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Finding Confidence Intervals

• (Ex 7) After sampling 100 cars on the I-90, Joe
  found a point estimate speed 61 mph and a
  standard deviation of 7.2 mph. He wishes to
  construct a 99 confidence interval for this
  data.

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Finding an appropriate sample size

• This will be used to achieve a specific
  confidence level for your study.

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Find a sample size

• (Ex 8) Bob wants to get a more accurate idea of
  the average on Stats Test 3 of all NCCC stats
  class students . How large of a sample will he
  need to be within 2 percentage points (margin of
  error), at a 95 confidence level, assuming we
  know the s 9.4?

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How did we get this?
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Finish Bobs Study

• Ex (9) - Now lets say Bob wants to perform his
  study, finds the point of estimate for Test 3
  83, with a SD of 9.4 and confidence level of 95.
  Find the confidence interval for this study.

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What about an interval found with a small sample
size? (chp 6.2)

• To do these problems we will need
• TABLE 5 t-Distribution.
• Determine from the problem n, x, s.
• Sample, mean, sample standard deviation.
• Use the MoE formula for small samples

t-value from Table 5
d.f. n-1 (degrees of freedom)
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Small Sample Confidence Int.

• (Ex 10) Trying to determine the class average
  for Test 3, Janet asks 5 students their grade
  on the test. She found a mean of 78 with a s
  7.6. Construct a confidence interval for her data
  at a 90 confidence level.

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What did we do?
d.f. 5-1 4 .90 Lc 2.132
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Or with TI-83/84

• STAT
• TESTS
• 8TInterval
• Stats
• Input each value, hit calculate.