Session 13 Notes, Section 7-3

Confidence Intervals for Mean with Small Samples

Estimation is an important aspect of inferential statistics.  Estimation is the process of estimating the value of a parameter (the population) from information obtained from a sample. In this session we will look at ways of determining how good our estimation probably is for samples larger less than 30.

At the end of this session you will be able to:

• Estimator
• Find the confidence interval for the mean when σ is unknown and n < 30.
  • Example
  • Example
• Your Turn:  Summary

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Estimator

• Three Properties of a Good Estimator
  • The estimator should be an unbiased estimator.
    • The expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.
  • The estimator should be consistent.
    • For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.
  • The estimator should be a relatively efficient estimator
    • Of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.
• Point and Interval Estimates
  • A point estimate is a specific numerical value of a parameter.
    • The best point estimate of the population mean, µ, is the sample mean, X^bar.
  • An interval estimate of a parameter is an interval or a range of values used to estimate the parameter.
    •  This estimate may or may not contain the value of the parameter being estimated.

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Find the confidence interval for the mean when σ is unknown and n < 30.

• t Distribution or Student t Distribution
  • When the population sample size is less than 30, and the standard deviation is unknown, the t distribution must be used.
• Characteristics of the t Distribution or Student t Distribution
  • The t Distribution or Student t Distribution is similar to the standard normal distribution in the following ways:
    • It is bell shaped.
    • It is symmetrical about the mean.
    • The mean, median, and mode are equal to 0 and are located at the center of the distribution.
    • The curve never touches the x axis
  • The t Distribution or Student t Distribution differs from the standard normal distribution in the following ways.
    • The variance is greater than 1.
    • The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to sample size.
    • As the sample size increases, the t distribution approaches the standard normal distribution.
• t Distribution or Student t Distribution and degrees of freedom
  • The degrees of freedom are the number of values that are free to vary after a sample statistic has been computed.
  • The degrees of freedom for the confidence interval for the mean are found by subtracting 1 from the sample size. That is,
    • d.f. =  n - 1
• Formula for specific confidence interval for the mean when σ is unknown and n < 30
  • X - t_α/2 (s/√n) < μ < X + t_α/2 (s/√n) 
• When to Use the z or t Distribution
  • Figure 7-8 on page 343 presents a nice chart for determining when to use the z-distribution or t-distribution (student t-distribution).

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Example.

Find the 95% confidence interval for the following sample:
625, 675, 535, 406, 512, 680, 483, 522, 619, 575

Use the formula  X - t_α/2 (s/√n) < μ < X + t_α/2 (s/√n)
 

Confidence interval - mean
95% confidence level
563.2 mean
87.9 std. dev.
10 n
2.262 t (df = 9)
62.880 half-width
626.080 upper confidence limit
500.320 lower confidence limit

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Example.

Twenty randomly selected automobiles were stopped, and the tread depth of the right front tire was measured. The mean was 0.35 inch, and the standard deviation was 0.07 inch. Find the 95% confidence interval of the mean depth. Assume the variable is approximately normally distributed.

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Summary

• A good estimator must be unbiased, consistent, and relatively efficient.
• There are two types of estimates of a parameter: point estimates and interval estimates.
• A point estimate is a single value. The problem with point estimates is that the accuracy of the estimate cannot be determined, so the interval estimate is preferred.
• By calculating a 90%, 95% or 99% confidence interval about the sample value, statisticians can be 90%, 95% or 99% confident that their estimate contains the true parameter.
• When the sample is n < 30, the confidence interval is calculated using

  X - t_α/2 (s/√n) < μ < X + t_α/2 (s/√n)

• Once the confidence interval of the mean is calculated, the z or t values are used depending on the sample size and whether the standard deviation is known.

More Examples:

Find the  t_α/2 value for the 98% confidence interval for the mean when n = 19 2.552

The average hemoglobin reading for a sample of 23 teachers was 12 grams per 100 milliliters, with a sample standard deviation of 3 grams. Find the 99% confidence interval of the true mean. Assume the variable is approximately normally distributed.  The true mean is between 10 and 14 grams per 100 milliliters based on this sample of 23 teachers.
13 women had an average heart rate of 117 beats per minute. The standard deviation of the sample was 7 beats. Find the 99% confidence interval of the true mean for the women. Please round to three decimal places and final answer to the nearest beat.  The true mean is between 111 and 123 beats per minute

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