Confidence Intervals about a Population Mean, s Known Section 8'1

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Confidence Intervals about a Population Mean, s
KnownSection 8.1

• Alan Craig
• 770-274-5242
• acraig_at_gpc.edu

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Objectives 8.1

• Compute the point estimate of m
• Compute confidence intervals about m with s known
• Understand the role of margin of error in
  constructing confidence intervals
• Determine sample size necessary for estimating
  the population mean

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From a Gallup Poll

• Survey Methods
• Results are based on telephone interviews with
  1,010 national adults, aged 18 and older,
  conducted March 2-4, 2007. For results based on
  the total sample of national adults, one can say
  with 95 confidence that the maximum margin of
  sampling error is 3 percentage points. In
  addition to sampling error, question wording and
  practical difficulties in conducting surveys can
  introduce error or bias into the findings of
  public opinion polls.

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Point Estimate

• A point estimate of a parameter is the value of a
  statistic that estimates the value of the
  parameter
• For example, x is a point estimate of the
  population mean m
• The sample standard deviation, s, is a point
  estimate of the population standard deviation s

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Characteristics ofBest Point Estimators

• A statistic is an unbiased estimator provided its
  expected value is equal to the value of the
  parameter. The sample mean, x is an unbiased
  estimator of the population mean m.
• The sample mean provides consistent estimates of
  the population mean. (The larger the sample, the
  closer x is to m.)
• Efficiency means that in repeated samples, a
  majority of the sample means will be close to
  the value of m.

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Best Point Estimate

• The sample mean, x, is the best point estimate of
  the population mean, m

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Computing a Point Estimate for m

• 16 (a) p. 350.
• Use the data below to compute a point estimate
  for the population mean flight time between
  Albuquerque and Dallas on American Airlines.
• 117, 95, 109, 103, 111, 91, 100, 99, 106

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Computing a Point Estimate for m

• 16 (a) p. 350.
• Use the data below to compute a point estimate
  for the population mean flight time between
  Albuquerque and Dallas on American Airlines.
• x (117951091031119110099106) / 9
• 103.4

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Confidence Intervals

• In the point estimate that we just found, there
  is probably some sampling error.
• This is because the sample does not have all the
  information from the population. Different
  samples probably have different sample means.
• We might give an estimated range instead of a
  point. e.g., 103.4 3 minutes
• The 3 minutes is the margin of error
• Further, we might want to state how confident we
  are that the given range includes the population
  mean 90, 95, 99

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Confidence Intervals
x - E x x E

• A confidence interval is an estimated interval
  around a statistic with a given confidence level
  (probability) that the interval contains the
  parameter.

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Confidence Intervals

• A confidence interval estimate of a parameter
  consists of an interval of numbers, along with a
  measure of the likelihood that the interval
  contains the unknown parameter.
• The level of confidence in a confidence interval
  is the proportion of intervals that will contain
  m if a large number of repeated samples is
  obtained. The level of confidence is denoted
  (1-a) 100.
• If a .05, then level of confidence is 95

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Confidence Intervals

• We will construct confidence intervals around the
  population mean, m, using the fact that the
  random variable x is (at least approximately)
  normally distributed
• Assumes the population is normal or the sample
  size 30!
• x has mean m and standard deviation

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Confidence Intervals

• From Table II, we know that 95 of the values of
  a normally distributed random variable are within
  1.96 standard deviations of the mean. For x,

Distribution of x
.025 in each tail
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Confidence Intervals

• Because 95 of the values of a normally
  distributed random variable are within 1.96
  standard deviations of the mean, we know that

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Confidence Intervals

• The 95 confidence interval is often written as
• Point Estimate Margin of Error

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Confidence Intervals

• Look at Table 3 on p. 342. In Table 3, 95
  confidence intervals have been constructed for
  each of 20 samples.
• Normal population
• m 100, s 16, n 15.
• Lets check the calculation for Sample 20

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Confidence Intervals

• Look at Table 3 on p. 342.
• Normal population, m 100, s 16, n 15.
• Lets check the calculation for Sample 20.
• The sample mean, x 102.28

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Confidence Intervals

• Look at Table 3 on p. 342.
• Normal population, m 100, s 16, n 15.
• Lets check the calculation for Sample 20.
• The sample mean, x 102.28

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Confidence Intervals

• Lets generate confidence intervals from the
  books website to see the concept in action
• Chapter 8
• Statlets
• Another interesting site at Rice U.

http//wps.prenhall.com/esm_sullivan_funstats_1/
http//www.ruf.rice.edu/lane/stat_sim/conf_interv
al/
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Critical Values

• Critical Values
• Where represents the Z-score that
  separates an area of a/2 in the right tail from
  the rest of the distribution in a (1-a) 100
  confidence interval.

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Critical Values

• Some critical values commonly used in
  constructing confidence intervals

Critical Value
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Interpreting Confidence Intervals

• A (1-a) 100 confidence interval means that if
  we obtained many simple random samples of size n
  from the population whose mean m is unknown, then
  approximately (1-a) 100 of the intervals would
  contain m.

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Interpreting Confidence Intervals

• A 95 confidence interval with a sample mean of
  103.4 and a margin of error of 3 minutes
• We are 95 confident that the population mean is
  between 100.4 and 106.4.
• Caution We are NOT saying that there is a 95
  probability that m lies between 100.4 and 106.4.

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Interpreting Confidence Intervals

• The level of confidence refers to the method used
  to find the confidence interval, not the interval
  itself.
• We are saying that the method we are using to
  compute the confidence interval should result in
  an interval that contains the population mean, m,
  (1-a) 100 of the time.

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Constructing (1-a)100 Confidence Intervals
about m, with s Known

• Suppose a simple random sample of size n is taken
  from a population with unknown mean, m, and known
  standard deviation s. A (1-a) 100 confidence
  interval for m is given by
• Note n 30 OR population normally distributed

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Z-intervals

• Because the confidence interval with s known uses
  Z-scores, it is sometimes called a Z-interval.
• In particular, this is what the TI calculator
  calls it.

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Constructing a Z-interval

• 16 (b) p. 350.
• n 9, so we need to check for normality using a
  normal probability plot and a box plot to check
  for outliers.
• Does this data appear to be normally distributed?

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Constructing a Z-interval

• 16 (b) p. 350.
• n 9, so we need to check for normality using a
  normal probability plot and a box plot to check
  for outliers.
• Does this data appear to be normally distributed?
  YES

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Constructing a Z-interval

• 16 (c) p. 350.
• Construct and interpret a 95 confidence interval
  for mean flight time. Recall that from (a), the
  sample mean was 103.4. Also s 8, n 9.

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Constructing a Z-interval

• 16 (c) p. 350.
• Construct and interpret a 95 confidence interval
  for mean flight time. Recall that from (a), the
  sample mean was 103.4. Also s 8, n 9. (Using
  Table II)

31
Constructing a Z-interval

• 16 (c) p. 350.
• Construct and interpret a 95 confidence interval
  for mean flight time. Recall that from (a), the
  sample mean was 103.4. Also s 8, n 9. (Using
  Calculator)

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Constructing a Z-interval

• 16 (c) p. 350.
• Place data into L1.
• Go to STAT?TESTS?7 ZInterval
• Select Inpt Data, s 8, List L1, Freq 1,
  C-Level .95, highlight Calculate, then ENTER

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Constructing a Z-interval

• 16 (c) p. 350.
• Construct and interpret a 95 confidence interval
  for mean flight time. Recall that from (a), the
  sample mean was 103.4. Also s 8, n 9. (Using
  Calculator)
• How does this compare to the values calculated
  using Table II?

Lower and Upper Bounds
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Constructing a Z-interval

• 16 (d) p. 350.
• Construct and interpret a 90 confidence interval
  for mean flight time. Recall that from (a), the
  sample mean was 103.4. Also s 8, n 9. (Using
  Calculator)

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Constructing a Z-interval

• 16 (d) p. 350.
• Construct and interpret a 90 confidence interval
  for mean flight time. Recall that from (a), the
  sample mean was 103.4. Also s 8, n 9. (Using
  Calculator)

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Margin of Error

• The margin of error, E, in a (1-a) 100
  confidence interval in which s is known is given
  by
• where n is the sample size.
• Note The population must be normally distributed
  or n 30

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Margin of Error

• Note that the margin of error depends on the
• Level of confidence
• Standard deviation of the population,
• Sample size, n

Which of these can we control?
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Margin of Error and Confidence Levels

• How does increasing the level of confidence
  affect the margin of error?
• Simple random sample, n 45, s 3.8, x 59.2
• Compute E for a 90 confidence interval
• Compute E for a 98 confidence interval

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Margin of Error Example

• If level of confidence increases, margin of error
  increases.

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Margin of Error and Sample Size

• How does increasing the level of confidence
  affect the margin of error?
• Simple random sample, 90 confidence interval, s
  3.8, x 59.2
• Compute E for sample size, n 45
• Compute E for sample size, n 55

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Margin of Error Example

• If sample size increases, margin of error
  decreases.

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Determining Sample Size n

• The sample size required to estimate the
  population mean, m, with a level of confidence
• (1-a) 100 with a specified margin of error, E,
  is given by
• where n is rounded up to the nearest whole number.

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Example 28, p. 354

• You want to estimate mean serum HDL cholesterol
  of all 20-29 year-old males.
• How many subjects would be needed in order to
  estimate the mean serum HDL cholesterol of all
  20-29-year-old males within 1.5 points with 90
  confidence, assuming that s 12.5?
• How many for 98 confidence?
• How does the increase in confidence affect the
  sample size required?

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Determining Sample Size n

• Estimate the mean within 1.5 points with 90
  confidence, assuming that s 12.5?
• Remember to always round up

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Determining Sample Size n

• Estimate the mean within 1.5 points with 98
  confidence, assuming that s 12.5?
• Remember to always round up

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Questions

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