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ESTIMATES AND SAMPLE SIZES WITH ONE SAMPLE

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Title: ESTIMATES AND SAMPLE SIZES WITH ONE SAMPLE


1
ESTIMATES AND SAMPLE SIZES WITH ONE SAMPLE
2
Overview
3
Applications of Inferential Statistics
  • Estimate the value of a population parameter.
  • Test some claim (or hypothesis) about a
    population.

4
Estimating a Population Proportion
5
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) .

6
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.

7
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.

8
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.

9
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.

10
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.)

11
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.

12
Critical Values in the Standard Normal
Distribution

13
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.

14
Example
  • Find the critical values for the following
    confidence levels
  • 90
  • 95
  • 99

15
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

16
Confidence Interval (or Interval Estimate) for
the Population Proportion p
  • whereThe
    confidence interval is often expressed in the
    following equivalent formats.or

17
Round-Off Rule for Confidence Interval Estimates
of p
  • Round the confidence interval limits for p to
    three significant digits.

18
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.

19
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

20
Procedure for Constructing a Confidence Interval
for p (continued)
  • Round the resulting confidence interval limits to
    three significant digits.

21
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.

22
Interpreting a Confidence Interval
  • We are 95 (or 90 or 95) confident that the
    interval actually does contain
    the true value of p.

23
Determining Sample Size
  • How large should our sample size be if we want
    the margin of error to be less than some given
    value?

24
Sample Size for Estimating Proportion p
  • When an estimate of is known
  • When no estimate of is known

25
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.

26
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.

27
Estimating a Population Mean Known
28
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

29
Point Estimate of
  • The sample mean is the best point estimate of
    the population mean .

30
Confidence Interval Estimate of the Population
Mean (With Known)
  • whereoror

31
Definition
  • The two values and are called
    confidence limits.

32
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
    .

33
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.

34
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.

35
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.

36
Sample Size for Estimating Mean
  • where critical z score based on the
    desired confidence levelE desired
    margin of error population standard
    deviation

37
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.

38
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.

39
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.

40
Estimating a Population Mean Not Known
41
Requirements for Estimating when is Unknown
  • The sample is a simple random sample.
  • Either the sample is from a normally distributed
    population or

42
Point Estimate of
  • The sample mean is the best point estimate of
    the population mean .

43
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 .

44
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.

45
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

46
Student t Distribution

47
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

48
Margin of Error E for the Estimate of (With
Not Known)
  • where has n 1 degrees of freedom.

49
Confidence Interval for the Estimate of the
(With Not Known)
  • whereoror

50
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 .

51
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.

52
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.
53
Choosing the Appropriate Distribution
54
Choosing the Appropriate Distribution

55
Estimating a Population Variance
56
Requirements for Estimating or
  • The sample is a simple random sample.
  • The population MUST have normally distributed
    values (even if the sample is large).

57
Chi-Square Distribution
  • where n sample size
    s2 sample variance
    population variance degrees of
    freedom n - 1

58
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.

59
Chi-Square Distribution

60
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.

61
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.

62
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

63
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).

64
Confidence Interval (or Interval Estimate) for
the Population Variance
  • To find the confidence interval for the
    population standard deviation , use

65
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

66
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.

67
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.

68
Determining Sample Size

69
Example
  • Find the minimum sample size needed to be 99
    confident that the IAA sample standard deviation
    is within 20 of .
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