ESTIMATES AND SAMPLE SIZES WITH ONE SAMPLE

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ESTIMATES AND SAMPLE SIZES WITH ONE SAMPLE
2
Overview
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Applications of Inferential Statistics

• Estimate the value of a population parameter.
• Test some claim (or hypothesis) about a
  population.

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Estimating a Population Proportion
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Objective

• Given a sample statistic, estimate the value of
  the population parameter (for this section, we
  will use the sample proportion to estimate the
  value the population proportion p) .

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Requirements for Using a Normal Distribution as
an Approximation to a Binomial Distribution

• The sample is a random sample.
• The conditions for the binomial distribution are
  satisfied. That is, there is a fixed number of
  trials, the trials are independent, there are two
  categories of outcomes, and the probabilities
  remain constant for each trial.
• The normal distribution can be used to
  approximate the distribution of sample
  proportions because and
  are both satisfied. Because p and q are unknown,
  we use the sample proportion to estimate their
  values.

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Notation for Proportions

• p proportion of successes in the entire
  population
• sample proportion of x successes in
  a sample of size n.
• sample proportion of failures in
  a sample of size n.

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Definition

• A point estimate is a single value (or point)
  used to approximate a population parameter.
• The sample proportion is the best point
  estimate of the population proportion p.

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Definition

• A confidence interval (or interval estimate) is a
  range (or an interval) of values used to estimate
  the true value of a population parameter. A
  confidence interval is sometimes abbreviated as
  CI.

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Definition

• The confidence level is the probability (often
  expressed as the equivalent percentage value,
  such as 95) that is the proportion of times that
  the confidence interval actually does contain the
  population parameter, assuming that the
  estimation process is repeated a large number of
  times. (the confidence level is also called the
  degree of confidence, or the confidence
  coefficient.)

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Notation for Critical Value

• The critical value is the positive z value
  that is at the boundary separating an area of
  in the right tail of the standard normal
  distribution. (The value of is at the
  vertical boundary for the area of in the
  left tail.) The subscript is simply a
  reminder that the z score separates an area of
  in the right tail of the standard normal
  distribution.

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Critical Values in the Standard Normal
Distribution

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Definition

• A critical value is the number on the borderline
  separating sample statistics that are likely to
  occur from those that are unlikely to occur. The
  number is a critical value that is a z
  score with the property that it separates an area
  ofin the right tail of the standard normal
  distribution.

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Example

• Find the critical values for the following
  confidence levels
• 90
• 95
• 99

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Definition

• When data from a simple random sample are used to
  estimate a population proportion p, the margin of
  error, denoted by E, is the maximum likely (with
  probability ) difference between the
  observed sample proportion and the true
  value of the population proportion p. The margin
  of error E is also called the maximum error of
  the estimate and can be found as follows

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Confidence Interval (or Interval Estimate) for
the Population Proportion p

• whereThe
  confidence interval is often expressed in the
  following equivalent formats.or

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Round-Off Rule for Confidence Interval Estimates
of p

• Round the confidence interval limits for p to
  three significant digits.

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Procedure for Constructing a Confidence Interval
for p

• Check that the requirements for this procedure
  are satisfied. (For this procedure, check that
  the sample is a simple random sample, the
  conditions for the binomial distribution are
  satisfied, and the normal distribution can be
  used to approximate the distribution of sample
  proportions because and
  are both satisfied.)
• Refer to Table A-2 and find the critical
  valuethat corresponds to the desired confidence
  level.

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Procedure for Constructing a Confidence Interval
for p (continued)

• Evaluate the margin of error
• Using the value of the calculated margin of error
  E and the value of the sample proportion ,
  find the values of and .
  Substitute those values in the general format for
  the confidence intervaloror

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Procedure for Constructing a Confidence Interval
for p (continued)

• Round the resulting confidence interval limits to
  three significant digits.

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Example

• As part of the National Health and Nutrition
  Examination Survey, iron levels were checked for
  a sample of 786 girls aged 12 to 15. Iron
  deficiency was detected in 71 of those sampled.
  Find a 95 confidence interval estimate of the
  population proportion p.

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Interpreting a Confidence Interval

• We are 95 (or 90 or 95) confident that the
  interval actually does contain
  the true value of p.

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Determining Sample Size

• How large should our sample size be if we want
  the margin of error to be less than some given
  value?

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Sample Size for Estimating Proportion p

• When an estimate of is known
• When no estimate of is known

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Round-Off Rule for Determining Sample Size

• In order to ensure that the required size is at
  least as large as it should be, if the computed
  sample size is not a whole number, round it up to
  the next higher whole number.

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Example

• What sample size would be needed to estimate the
  proportion of girls aged 12 to 15 with iron
  deficiency if the researcher wants 99 confidence
  that the sample proportion is in error by no more
  than 0.03? Use the sample proportion as a known
  estimate.

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Estimating a Population Mean Known
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Requirements for Estimating when is known

• The sample is a simple random sample.
• The value of the population standard deviation
  is known.
• Either or both of these conditions are satisfied
• The population is normally distributed, or

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Point Estimate of

• The sample mean is the best point estimate of
  the population mean .

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Confidence Interval Estimate of the Population
Mean (With Known)

• whereoror
  

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Definition

• The two values and are called
  confidence limits.

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Procedure for Constructing a Confidence Interval
for (with Known )

• Check that the requirements are satisfied.
  (Requirements We have a simple random sample,
  is known, and either the population appears to
  be normally distributed or .)
• Refer to Table A-2 and find the critical
  valuethat corresponds to the desired confidence
  level.
• Evaluate the margin of error
  .

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Procedure for Constructing a Confidence Interval
for (with Known ) (continued)

• Using the value of the calculated margin of error
  E and the value of the sample mean , find the
  values of and . Substitute those
  values in the general format for the confidence
  intervaloror
• Round the resulting values by using the following
  round-off rule.

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Round-Off Rule for Confidence Intervals Used to
Estimate

• When using the original set of data to construct
  a confidence interval, round the confidence
  interval limits to one more decimal place than is
  used for the original set of data.
• When the original set of data is unknown and only
  the summary statistics are used,
  round the confidence interval limits to the same
  number of decimal places used for the sample mean.

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Example

• The health of the bear population in Yellowstone
  National Park is monitored by periodic
  measurements taken from anesthetized bears. A
  sample of 54 bears has a mean weight of 182.9 lb.
  Assuming that is known to be 121.8 lb, find
  a 99 confidence interval estimate of the mean of
  the population of all such bear weights.

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Sample Size for Estimating Mean

• where critical z score based on the
  desired confidence levelE desired
  margin of error population standard
  deviation

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Round-Off Rule for Sample Size n

• When finding the sample size n, if the use
  ofdoes not result in a whole number, always
  increase the value of n to the next larger whole
  number.

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Dealing with Unknown When Finding Sample Size

• Use the range rule of thumb to estimate the
  standard deviation as follows
• Conduct a pilot study be starting the sampling
  process. Based on the first collection of at
  least 31 randomly selected sample values,
  calculate the sample standard deviation s and use
  it in place of . The estimated value can
  then be improved as more sample data are
  obtained.
• Estimate the value of by using the results of
  some other study that was done earlier.

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Example

• We are still very concerned about the health of
  the bear population in Yellowstone National Park.
  How large a sample is needed if we want to be 90
  confident that the sample mean weight is within 5
  lb of the true population mean? Use the sample
  standard deviation from the previous study.

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Estimating a Population Mean Not Known
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Requirements for Estimating when is Unknown

• The sample is a simple random sample.
• Either the sample is from a normally distributed
  population or

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Point Estimate of

• The sample mean is the best point estimate of
  the population mean .

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Student t Distribution

• If the distribution of a population is
  essentially normal (approximately bell-shaped),
  then the distribution ofis essentially a
  Student t distribution for all samples of size n.
  The Student t distribution, often referred to
  simply as the t distribution, is used to find
  critical values denoted by .

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Important Properties of the Student t Distribution

• The Student t distribution is different for
  different sample sizes.
• The Student t distribution has the same general
  symmetric bell shape as the standard normal
  distribution, but it reflects the greater
  variability (with wider distributions) that is
  expected with small samples.
• The Student t distribution has a mean of t 0
  (just as the standard normal distribution has a
  mean of z 0).
• The standard deviation of the Student t
  distribution varies with the sample size, but it
  is greater than 1 (unlike the standard normal
  distribution, which has ).
• As the sample size n gets larger, the Student t
  distribution gets closer to the standard normal
  distribution.

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Definition

• The number of degrees of freedom for a collection
  of sample data is the number of sample values
  that can vary after certain restrictions have
  been imposed on all data values.
  degrees of freedom n - 1

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Student t Distribution

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Example

• A sample size of n 20 is a simple random sample
  selected from a normally distribution population.
  Find the critical values for the following
  confidence levels
• 90
• 95

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Margin of Error E for the Estimate of (With
Not Known)

• where has n 1 degrees of freedom.

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Confidence Interval for the Estimate of the
(With Not Known)

• whereoror
  

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Procedure for Constructing a Confidence Interval
for (with Not Known)

• Check that the requirements are satisfied.
  (Requirements We have a simple random sample,
  and either the population appears to be normally
  distributed or .)
• Using n 1 degrees of freedom, refer to Table
  A-3 and find the critical value that
  corresponds to the desired confidence level.
• Evaluate the margin of error .

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Procedure for Constructing a Confidence Interval
for (with Not Known) (continued)

• Using the value of the calculated margin of error
  E and the value of the sample mean , find the
  values of and . Substitute
  those values in the general format for the
  confidence intervaloror
• Round the resulting confidence interval limits.

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Example

• As part of a study on plant growth, a plan
  physiologist grew 13 individually potted soybean
  seedlings of the type called Wells II. She raised
  the plants in a greenhouse under identical
  environmental conditions (light, temperature,
  soil, etc.). She measured the total stem length
  (cm) for each plant after 16 days of growth.

Assuming that the distribution of lengths is
approximately normal, calculate a 95 confidence
interval for the mean stem length.
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Choosing the Appropriate Distribution
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Choosing the Appropriate Distribution

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Estimating a Population Variance
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Requirements for Estimating or

• The sample is a simple random sample.
• The population MUST have normally distributed
  values (even if the sample is large).

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Chi-Square Distribution

• where n sample size
  s2 sample variance
  population variance degrees of
  freedom n - 1

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Properties of the Distribution of the Chi-Square
Statistic

• The chi-square distribution is not symmetric,
  unlike the normal and Student t distributions.
  (As the number of degrees of freedom increases,
  the distributions becomes more symmetric.)
• The values of chi-square can be zero or positive,
  but they cannot be negative.
• The chi-square distribution is different for each
  number of degrees of freedom, and the number of
  degrees of freedom is given by df n 1 in this
  section. As the degrees of freedom increases, the
  chi-square distribution approaches a normal
  distribution.

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Chi-Square Distribution

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Critical Values of the Chi-Square Distribution

• In Table A-4, each critical value of corresponds
  to an area given in the top row of the table, and
  that area represents the total region located to
  the right of the critical value.

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Notation

• With the total area of divided equally
  between the two tails of a chi-square
  distribution, denotes the left-tailed
  critical value and denotes the right-tailed
  critical value.

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Example

• A sample size of n 20 is a simple random sample
  selected from a normally distribution population.
  Find the critical values and for the
  following confidence levels
• 90
• 95

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Estimators of and

• The sample variance s2 is used as the best point
  estimate of the population variance .
• The sample standard deviation s is commonly used
  as a point estimate of (even though it is a
  biased estimator).

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Confidence Interval (or Interval Estimate) for
the Population Variance

• To find the confidence interval for the
  population standard deviation , use

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Procedure for Constructing a Confidence Interval
for or

• Check that the requirements for the methods of
  this section are satisfied. (Requirements The
  sample is a simple random sample and a histogram
  or normal quantile plot suggests that the
  population has a distribution that is very close
  to a normal distribution.)
• Using n 1 degrees of freedom, refer to Table
  A-4 and find the critical values and
  that correspond to the desired confidence level.
• Evaluate the upper and lower confidence interval
  limits using this format of the confidence
  interval

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Procedure for Constructing a Confidence Interval
for or(continued)

• If a confidence interval estimate of is
  desired, take the square root of the upper and
  lower confidence interval limits and change
  to .
• Round the resulting confidence interval limits.
  If using the original set of data, round to one
  more decimal place than is used for the original
  set of data. If using the sample standard
  deviation or variance, round the confidence
  interval limits to the same number of decimal
  places.

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Example

• In a study of the effectiveness of a gluten-free
  diet in first-degree relatives of patients with
  Type I diabetes, researchers placed seven
  subjects on a gluten-free diet for 12 months.
  Prior to the diet, they took a baseline
  measurement of the diabetes related insulin
  autoantibody (IAA). The seven subjects had IAA
  units of 9.7, 12.3, 11.2, 5.1, 24.8,
  14.8, and 17.7Assuming that sample is a simple
  random sample and that the sample data appear to
  come from a population with a normal
  distribution, find a 95 confidence interval
  estimate of the population standard deviation.

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Determining Sample Size

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Example

• Find the minimum sample size needed to be 99
  confident that the IAA sample standard deviation
  is within 20 of .