Interval estimates of the mean for small n, s unknown. Estimates of sample size.

1
Interval estimates of the mean for small n, s
unknown. Estimates of sample size.

• ASW, 8.2 8.4

Economics 224 Notes for October 15
2
Interval estimates using the t distribution

• When a random sample of size n is drawn from a
  normally distributed population whose standard
  deviation s is unknown, the sampling distribution
  of the sample mean has a t distribution.
• The interval estimate has the same format as
  earlier, but with a t value replacing the Z
  value, and the sample standard deviation s
  replacing s. The interval estimate of the
  population mean µ is
• A confidence level must be specified for each
  interval estimate the confidence level and the
  sample size determine the t value.

3
t distribution

• The shape of the t distribution is similar to
  that of the normal distribution bell-shaped,
  peaked in the centre, symmetrical about the mean,
  and asymptotic to the horizontal axis.
• The table of the t distribution is a standardized
  t distribution, that is, the t values have mean 0
  and standard deviation 1.
• There are many different t distributions, one for
  each degree of freedom (see explanation on a
  later slide).
• For small degrees of freedom, the t distribution
  is very dispersed. As the degrees of freedom
  increase, the t distribution becomes more
  concentrated around the mean.
• The limiting distribution for the t distribution
  is the normal distribution. That is, as the
  degrees of freedom increase, the t distribution
  is approximated by the normal distribution.

4
The concept of degrees of freedom (df)

• Degrees of freedom refers to how many sample
  values can vary freely. In many statistical
  procedures, some sample values are constrained by
  the parameters to be estimated.
• When a t distribution describes the sampling
  distribution of the mean, one degree of freedom
  is lost since s is used as an estimate of s (ASW,
  304). In this case, the t distribution has n-1
  degrees of freedom, where n is the sample size.
• Degrees of freedom are used in the chi-square
  test and distribution and in regression and
  analysis of variance models. The type and number
  of constraints and degrees of freedom differ from
  model to model.

5
t table (ASW, 303 and Appendix B, Table 2

• In ASW, the t table gives areas in the upper tail
  of the distribution. The t distribution is
  symmetric about a mean of t 0 so the same area
  for the lower tail is given by the negative of
  the t value in the table.
• Each degree of freedom defines a different t
  distribution. For degrees of freedom above 100,
  use Z values from the standard normal
  distribution, since the normal distribution
  closely approximates the t distribution for large
  df.
• t values are given in the body of the table, with
  areas under the curve given at the top of each
  column, and df at the start of each row.
• The values in the table state the t value, or
  number of standard deviations, to the right of
  the centre of the distribution that is required
  to include all but the area in the right tail of
  the distribution.

6
General notation for interval estimates

• The confidence coefficient is given the symbol
  (1-a). a is the first letter of the Greek
  alphabet and is termed alpha. For interval
  estimates, a is merely a symbol used to denote
  the area in the two tails of a distribution. The
  area in the middle of the distribution is (1-a)
  or (1-a) x 100 and there is a/2 of the area in
  each of the tails of the distribution.
• When a sample mean is normally distributed, the Z
  values for the 95 interval estimate, or the (1
  0.05) x 100 95 interval, are 1.96. That is,
  if a 0.05,
  
• and the interval estimate of the population
  mean µ is

7
Notation for interval estimates using the t
distribution

• For the t distribution, the notation is the same
  as in the last slide, with t replacing Z. The
  only addition is that for each df, there is a
  different t value.
• For a with a t distribution. the t values for the
  (1a) interval are ta/2 for appropriate df.
• If a sample mean has a t distribution, df n-1,
  that is, the sample size minus 1.
• For a 98 interval estimate of a population mean,
  where the sample size is n 9, or df n 1
  8, t 2.896.
• In this case, the interval estimate of the
  population mean µ is

8
Example of wages of workers employed at new jobs
after a plant shutdown I

• Prior to the shutdown of an Ontario manufacturing
  plant in the 1990s, male were paid 13.76 per
  hour and females 11.80 per hour.
• Two years after the workers were laid off,
  researchers located some of these laid off
  workers. For twelve males workers who found new
  jobs, the mean hourly wage was 12.20, with a
  standard deviation of 3.27. For twelve female
  workers who found new jobs, the mean hourly wage
  was 8.11, with a standard deviation of 3.53.
• Obtain 90 interval estimates of the mean wage
  for all laid off male workers who found new jobs.
  For female workers. What do you conclude from
  these results?

Source Data and research from Belinda Leach and
Anthony Winson, Bringing Globalization Down to
Earth Restructuring and Labour in Rural
Communities, Canadian Review of Sociology and
Anthropology, 323, August 1995.
9
Example of wages II

• These are small samples, and for neither males
  nor females is the standard deviation of wages
  known. While male wages in the new jobs may not
  be exactly normally distributed, assume that they
  are symmetrically distributed or close to normal.
  Assume the same for the distribution of wages of
  female workers. Assume that each sample is a
  random sample of all laid off workers who had new
  jobs at the time of the study.
• From the above assumptions, for each of male and
  female workers who found new jobs, the
  distribution of sample mean pay is a t
  distribution. In each case, the sample has size
  n 12, so the df associated with each interval
  estimate is df 12 -1 11. For a 90 interval
  estimate, there is 10 or 0.10 of the area in the
  middle of the distribution and 5 or 0.05 of the
  area in each of the two tails of the
  distribution. The appropriate t value for 11 df
  and 90 confidence is 1.796.

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Example of wages III

• For males, let µ be the mean wage of all those
  laid off male workers who found new jobs. The
  90 interval estimate for the population mean µ
  is
• or 12.20 0.94 or (11.26 , 13.14).
• For females, the procedure is the same and the
  resulting 90 interval estimate is
• or 8.11 1.02 or (7.09 , 9.13).

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Example of wages IV

• These interval estimates provide reasonable
  certainty that the mean hourly pay of laid off
  workers has declined.
• For males, the 90 confidence interval is 11.30
  to 13.14, so the mean pay for all re-employed
  males is very likely below the pre-layoff level
  of 13.76.
• For females the situation appears worse. The 90
  interval estimate of mean pay of re-employed
  females is from 7.09 to 9.13, an interval well
  below the pre-layoff level of 11.80.
• Neither result is certain but evidence at the
  time of the study is that workers, especially
  females, experienced a decline in mean pay, as
  compared with the pre-layoff situation.
• Cautions samples may not be random,
  distribution of pay may not be normal,
  uncertainty with only 90 confidence.

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Using the t distribution

• Small n often occurs in practice.
• s unknown is the usual situation.
• Normal distribution of population. This is
  unlikely but so long as the population
  distribution is close to symmetric, this should
  not produce unreliable results.
• In these situations, when estimating a population
  mean, it is advisable to use the t distribution
  as the sampling distribution of the sample mean,
  rather than the normal distribution.
• For larger sample sizes, the sampling
  distribution of sample means can be approximated
  by the normal distribution.
• All of the above assume that the sample is a
  random sample, or is equivalent to a random
  sample.

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t distribution in economic analysis

• Much economic analysis uses very large sample
  sizes. But there are situations where n is small
  and the population standard deviation is unknown.
  Then the sample mean has a t distribution. When
  n gt100, the normal provides a good approximation
  to the t distribution.
• Experiments, administrative data, or other
  situations with a small number of cases.
• Measurement error is often close to normally
  distributed.
• Regression analysis, especially with time series
  data, where the number of observations across
  time is not large. Regression coefficients have
  a t distribution (ASW, Ch. 12).
• Economic variables are sometimes assumed to be
  normally distributed, but with unknown
  variability, so the t distribution is used for
  the distribution of the sample mean.

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Estimating sample size (ASW, 310-313)

• Prior to conducting a research study, it is often
  useful to estimate the sample size required to
  achieve a particular margin of error, specifying
  a confidence level.
• While this may not be the final sample size a
  researcher obtains, the following calculations
  provide an estimate of the number of population
  elements from which a researcher should attempt
  to obtain data. This, in turn, can be used to
  plan the research study and estimate the time and
  cost that will be required to conduct it.
• Cost may be too great, respondents may refuse to
  participate, nonresponse to some questions, time
  may be insufficient.
• The method examined here provides the required
  sample size for a random sample from a
  population, given the margin of error and
  confidence level.

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Formula

• Margin of error E.
• Confidence level (1 a) 100 and the
  corresponding normal value is Za/2.
• Population standard deviation is s.
• n is the required sample size when random
  sampling from this population.

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Rationale for formula

• Formula for interval estimate is
• Researcher wishes an interval
• Let
• And solve this expression for n, giving

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Example of sample size I

• A manager at Access Communications wants to know
  whether it is worthwhile to target university
  students for a promotion. In order to do this
  she would like to know how many minutes of TV
  students watch each day, accurate to within five
  minutes, with 99 confidence. The upper limit
  of budget expenditures for the study is 1,000.
• You have been hired as a consultant to the
  manager and your task is to conduct a sample of
  university students to obtain the required
  information. What sample size is required?
  What would you recommend to the manager?

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Example of sample size II

• Fortunately you have kept the Excel worksheet
  from Economics 224 and when you check it, you
  determine that the standard deviation of the
  hours students in Economics 224 reported watching
  TV daily was 1.298 hours.
• From this, you use the requirements specified by
  the manager, that is, a margin of error of 5
  minutes or E 5/60 0.0833 hours and 99
  confidence. The Z value is 2.576, so the
  required sample size is

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Example III

• You report that the required sample size is at
  least n 1,610.
• You also report that a larger sample size might
  be required, since you have an estimate of the
  variability of the population that may be low.
  That is, s for all university students might
  exceed 1.3 hours.
• If this is a random sample, to obtain a sampling
  frame and then contact each student by telephone,
  email, or Canada Post, you estimate that the cost
  of sampling is approximately 10 per student, for
  a total cost of well over 1,000.
• From this, you might recommend a smaller sample
  size, with relaxed requirements, say E 15
  minutes, 95 confidence, for a sample of around
  100 students. You might note that the required
  margin of error of five minutes is very difficult
  to obtain and too demanding.
• Explore less expensive methods of conducting the
  survey.

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Estimating s

• To obtain an estimate of the required n, some
  estimate of the population standard deviation s
  is required.
• Use s from previous studies or similar
  populations.
• Pilot study. Obtain a preliminary estimate of s.
• Judgment or best guess. Dividing the range by 4
  can produce a reasonable provisional estimate of
  s. If there are outliers, it may be best to
  eliminate these. For example, what is the s of
  income for Saskatchewan residents? Minimum 0
  and maximum might be 10 million plus. But make
  range from 0 to 100,000 and this may include 99
  plus of the population. Rough estimate of s for
  Saskatchewan income would be around 25,000.
  (For 2001, s 23,000, from Census).
• Structure sampling procedure so that the sample
  size can later be increased, if necessary.

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Additional notes about sample size I

• When obtaining n from the formula, round up.
• Make sure units for s and E in the formula are
  the same.
• n larger with
• Greater variability s in the population.
• Larger confidence level.
• Smaller margin of error E.
• Trade off between costs and accuracy of results.
• For a random sample, n does not depend on
  population size if the proportion of the
  population sampled (n/N) is small.
• It may not be possible to obtain the required n,
  so researcher will have to settle for a larger
  margin of error or reduced confidence level, or
  both. For example, time series data on Internet
  use may only be available for n 10 years.

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Additional notes about sample size II

• Sample size given by above formula indicates the
  number of population elements actually required
  in the study. If individuals or firms to be
  surveyed are reluctant to participate, cannot be
  found, or are unwilling to answer some questions,
  expand the required number of elements in the
  hope that the n indicated can be obtained. For
  example, if the formula indicates a required n
  500 and 25 nonresponse is expected, expand
  sample size to 650 or 700.
• Sampling procedure affects required sample size.
  Cluster samples might need to have larger n but
  stratification of a sample might reduce the
  required sample size. Different formulae for
  more complex sampling procedures.

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Weighting of sample elements

• This issue is not discussed in the text but is
  one that needs consideration in much survey
  sampling.
• Sampling procedure may be designed so each
  element selected in the sample represents a
  different number of population elements (eg.
  cluster, stratified, multistage sampling).
  Research methodology should report the weighting
  procedures to be used when conducting data
  analysis. Statistics Canada often includes a
  weight in the data set.
• Weighting may occur after data obtained, to
  estimate characteristics of population. For
  example, if males and females are about equal in
  number in a population but a sample has fewer
  males than females, data from males may be more
  heavily weighted when analyzing and reporting
  results.

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Conclusion about interval estimates and sample
size

• Formulae are precise but approximations are often
  used
• Random sample?
• Standard deviation of population?
• Confidence level arbitrary.
• Nonresponse and other nonsampling errors.
• When data come from samples, there is usually
  sampling error. Interval estimates and estimates
  of sample size are necessary but remember above
  cautions about their accuracy.
• Replication of studies, similar research on
  related topics and comparable populations.
• Careful sample and research design and data
  analysis.

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Later on Wednesday or on Monday

• Normal approximation to the binomial (ASW, 6.3).
• Sampling distribution of the sample proportion
  (ASW, 7.6).
• Interval estimate of a population proportion
  (ASW, 8.4).
• Sample size for estimation of a population
  proportion (ASW, 315-316).
• Review Monday during class and Tuesday at
  review session