Confidence Intervals

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MATH 1530 Elements of StatisticsDr. Kirsten Boyd

• Chapter 8
• Confidence Intervals

Slides adapted from Ms Smyth, Dr. Griffy and the
Weiss Text
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Sec. 8.1 Estimating a Population Mean

• If we want to use a statistic ( or s) to
  estimate a parameter (µ or s), can you be 100
  confident that your sample accurately represents
  the population?

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Estimating a Population Mean
point estimate for µ s point estimate
for s
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How Confident Can We Be?

• Can we assign a level of confidence to our point
  estimate?
• How certain are we that our statistic is a good
  estimator of the parameter?
• What is the probability that our sample
  accurately represents the population?

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Confidence-Interval
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Building a 95.44 CI

• 95.44 of data in a normal distribution are
  within two standard deviations of the mean
• Sampling distribution will be normal if original
  data is normal or if n is large
• Therefore, if original data is normal or if n is
  large, 95.44 of samples of size n will have a
  mean that is within two standard deviations of
  the population mean

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Building a 95.44 CI

• Recall
• Construct the interval consisting of all numbers
  that are within 2 standard deviations of the
  sample mean, as follows
• left endpoint of interval is x - 2(s / vn)
• right endpoint of interval is x 2(s / vn)
• NOTE For this process, you must be given the
  sample size, the sample mean and the population
  standard deviation

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95.44 CI

• If you take all possible sample means and their
  CIs, 95.44 of these CIs would contain µ
• Thus, there is a 95.44 chance that the CI
  contains µ

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Mobile Home Prices of 20 Random Samples(computer
simulation, see page 341)
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Problems 8.4 and 8.6 (page 342)

• The following list shows the number of baby
  snakes per litter, for 44 randomly-chosen litters
  of eastern cottonmouths.
• 5, 12, 7, 7, 6, 8, 12, 9, 7, 4, 9, 6, 12, 7, 5,
  6, 10, 3, 10, 8, 8, 12, 5, 6, 10, 11, 3, 8, 4, 5,
  7, 6, 11, 7, 6, 8, 8, 14, 8, 7, 11, 7, 5, 4
• 4.a. Use the data to obtain a point estimate
  for µ, the mean number of snakes per litter born
  to all eastern cottonmouths.
• 4.b. Is your answer to Part a. likely to equal
  µ exactly?
• 6. Assume that s, the population st. dev. for
  the number of snakes per litter, is 2.4.
• 6.a. Obtain an approx. 95.44 CI for the mean
  number of snakes per litter born to all eastern
  cottonmouths.
• 6.b. Interpret your answer to 6.a.
• 6.c. Why is the 95.44 CI that you obtained in
  Part a. not necessarily exact?

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Sec. 8.2 Confidence Intervals for One Population
Mean When s Is Known

• We dont want to be limited to just 95.44 CI!
• If the sampling distribution is normal, we can
  compute CI with any confidence level using
  z-scores.
• Recall from Sec. 6.2
• za means the z-score that has area a to its right
  (under the standard normal curve).
• Beware Table II and calculator both use areas to
  left.
• example
• z 0.025 invNorm(1-0.025) 1.96
• Or, use Table II to find z-score that has area
  0.975 (because 0.9751-0.025) to its left.

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Confidence Intervals When s is Known
95.44 of all samples have means within
of µ. An interval for any confidence
level can be calculated replacing 2 with za/2 ,
where a is obtained by subtracting the confidence
level from 1. example for 95.44 confidence, a
1-0.9544 0.0456 But now we can do this for
any confidence level.
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In Step 1, can also use calculator to find za/2
, using invNorm(1-a/2)
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Finding Alpha, a

• Lower case Greek letter alpha, a
• a 1 (level of confidence)
• Must write level of confidence as a number
  between 0 and 1, NOT as a percentage)
• Example for a 97 confidence interval,
• a 1 - 0.97 0.03

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Example Step 1

• For a level of confidence 95
• Determine a
• Determine a/2
• Determine za/2
• For a level of confidence 99
• Determine a
• Determine a/2
• Determine za/2

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Example Step 2

• Assume that for a certain sample of size 24, the
  mean of the x-values is 12. Suppose that we know
  that s 2.3.
• Part a. Find the C.I. for a 95 confidence
  level.
• Part b. Find the C.I. for a 99 confidence level.

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Guidelines for Using z-interval CI

• Use only when
• n lt 15, data is normal
• 15 n 30, data is approximately normal, or
• n 30

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Confidence Precision

• For a fixed sample size,
• lower CL better precision
• higher CL worse precision

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East German Prisoners PTSD

• Page 351, 8.35 (Apply procedure on page 345 to
  get a 95 CI)
• x number of months imprisoned
• Assume that s 42. For a sample of 32 patients
  with chronic PTSD, the mean duration of
  imprisonment was 33.4 months.
• Step 1 Find za/2
• Step 2 Find CI
• Step 3 Interpret the CI

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8.35 by Calculator
Using the calculator, you still must interpret
your results!

• Stat lt Tests lt 7 zInterval
• Highlight Stats lt Enter
• Fill in s, x, n, and C-Level (not )
• Highlight Calculate lt Enter

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Sec. 8.3 Margin of Error

• The margin of error is the part of the formula
  that is being added to and subtracted from x in
  calculating the CI.

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Meaning of margin of error

• The margin of error is the wiggle room around the
  sample mean
• If CI is from 3 (which is 5-2) to 7 (which is
  52), then the wiggle room is 2
• The margin of error is 2

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Visual Interpretation of Margin of Error
Width of CI CI max CI min 2E
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Precision
High Precision is the same thing as the CI
being short, so its the same as E being small
(since the length of the CI is 2E).
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Sample Size
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Prisoners PTSD

• Page 358, 8.65, continuation of 8.35 (Slide 20)
• x33.4, n32, s42
• Part a. Determine E, the margin of error.
• Part b. Explain what E means in terms of the
  accuracy of the estimate (in this context).
• Part c. Find the sample size needed to obtain a
  margin of error of 12 months and a 99 CL.
• Part d. Find a 99 CI for µ (the mean duration
  of imprisonment) if it is known that a certain
  sample of the size determined in Part c. has mean
  36.2 months.

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Sec. 8.4 Confidence Intervals for One Population
Mean When s Is Unknown
The studentized version of x is called t and is
like the standardized version z (shown at right),
except that it uses s (the sample st. dev.) in
place of s (the population st. dev.) So, use t
if you have only sample data and dont know s.
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Distribution of t is not normal, but the larger n
is, the closer it is to normal, because then s is
closer to s
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Key Fact
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t-scores (Table IV or calculator)

• Find the t-value that is larger than 95 of all
  t-values, for a sample size of 14. (Answer is
  t0.05 1.771)

On calculator 2nd gt VARS to get DISTR, then
option 4, invT(area to left, df) For example,
t0.05 invT(1-0.05,13)1.771
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When can we use the t-interval procedure?

• Only when x has a normal distribution, which
  happens when
• nlt 15, data is normal
• 15 n 30, data is approximately normal, or
• n 30
• If we have a sample of size less than 30 and we
  want to figure out whether we can use the
  t-interval procedure, we can use a normal
  probability plot (Sec. 6.4) to assess whether the
  data is normal.

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Example 8.9, page 363Dollar amounts lost in 25
pickpocket crimes
n 25, not a large enough sample to
automatically make the distribution of x normal.
Need to check whether x is approximately normal.
t-Interval Procedure is appropriate
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Example 8.10, page 364 Amount of chicken
consumed last year, by 17 randomly-chosen people
Sometimes a variable is normal only if you remove
outlier(s), which can be done only if theres a
good reason!
t-Interval Procedure is not appropriate
t-Interval Procedure is appropriate (sample size
is now 16)
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Page 368
8.82 For a t-curve with df8, find each t-value
and illustrate your results with a graph. Part
a. The t-value that has area 0.05 to its
right. Part b. t0.10 Part c. The t-value that
has area 0.01 to its left. (Hint if using
table a t-curve is symmetric about 0). Part d.
The two t-values that divide the area under the
curve into a middle area of 0.95 and two outer
areas of 0.025.
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Page 369
8.94 25 families of four that visited amusement
parks were selected. The cost of each visit
(rounded to the nearest dollar) is recorded
below. 156, 212, 218, 189, 172, 221, 175, 208,
152, 184, 209, 195, 207, 179, 181, 202, 166, 213,
221, 237, 130, 217, 161, 208, 220 (Note Using
the 1-Var Stats calculator function, you can
find the sample mean x193.32 and the sample st.
dev. s26.73.) Obtain and interpret a 95
confidence interval for the mean cost for a
family of 4 to spend the day at an amusement
park. Answer theres a 95 chance that the
average cost for a family of 4 to spend the day
at an amusement park is between 182.30 and
204.40.
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8.94 (previous slide), calculator method

• Enter data in a list (Stat lt Edit)
• Stat lt Tests lt 8TInterval
• Highlight Data, press Enter
• List name of list where you put the data
• Freq1
• C-Level 0.95
• Highlight Calculate, press Enter

If you use the calculator, you still must be able
to interpret your results!