Session 12 Notes, Sections 7-1 and 7-2

Confidence Intervals for Mean with Large 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 than 30.

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

• Estimator
• Find the confidence interval for the mean when σ is known or 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 (large samples) when σ is known or n ≥ 30.

• The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.
• A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.
• Formula for the Confidence Interval of the Mean for a Specific α (alpha - represents the total area in both tails of the standard normal distribution curve.  thus α/2 is the area under one tail)
  • X - z_α/2 (σ/√n) < μ < X + z_α/2 (σ/√n)
    • For a 90% confidence interval,   z_α/2 = 1.65 standard errors or standard deviations from the population mean.
    • For a 95% confidence interval,  z_α/2 = 1.96 standard errors or standard deviations from the population mean .
    • For a 99% confidence interval,  z_α/2 = 2.58 standard errors or standard deviations from the population mean.
• Maximum Error of Estimate
  • The maximum error of estimate is the maximum likely difference between the point estimate of a parameter and the actual value of the parameter. That is, The maximum error of estimate is the difference between the population mean μ and the point value X.

    The maximum error of estimate is represented by the term

    • z_α/2 (σ/√n)
      • (Note the relationship between the maximum error of estimate and the Confidence Interval)
      • (Note the relationship between the maximum error of estimate and the standard deviation)
      • (Note the relationship between the maximum error of estimate and the sample size)
      • (Note scientific calculator and computer calculations will deviate a bit from the calculations done using tables and a standard calculator due to rounding errors)
• Rounding Rule: In calculating the confidence interval using the sample mean and standard deviation, round to same number of decimal places as the mean.

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Example.

The president of a large university wishes to estimate the average age of the students presently enrolled. From past studies, the standard deviation is known to be 2 years. A sample of 60 students is selected, and the mean is found to be 23.3 years. Find the 95% confidence interval of the population mean. Please round your answers to one decimal place.

Since the 95% confidence level is desired, z sub alpha/2 = 1.96. Hence, substituting in the formula

X - z_α/2 (σ/√n) < μ < X + z_α/2 (σ/√n)

you get: 23.3 - 1.96 (2/(60^.5))  <  µ < 23.3 + 1.96 (2/(60^.5))
OR
23.3 - 0.5 <  µ < 23.3 + 0.5
OR
22.8 <  µ < 23.8
Hence, the president can say, with 95% confidence, that the average age of the student is between 22.8 and 23.8 years, based on 60 students.

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Example.

A survey of 50 adults found that the mean age of a person's primary vehicle is 5.9 years. Assuming the standard deviation of the population is 0.9 year, find the 99% confidence interval of the population mean.

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Example.

For 43, 44, 52, 42, 18, 41, 20, 41, 25, 53, 45, 22, 43, 25, 21, 23, 42, 21, 32, 27, 24, 33, 32, 36, 19, 47, 25, 19, 26, 20
find the 95% confidence Interval for the population mean when the standard deviation of the population is known as 11.

Confidence interval - mean
95% confidence level
32.03 mean
11 std. dev.
30 n
1.960 z
3.936 half-width
35.966 upper confidence limit
28.094 lower confidence limit
 

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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.
• Interval estimates with confidence intervals are used  frequently.
• When the sample is 30 ≤ n, the maximum error of estimate is represented by the term z_α/2 (σ/√n)
• 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 30 ≤ n, the confidence interval is calculated using the maximum error of estimate and

  X - z_α/2 (σ/√n) < μ < X + z_α/2 (σ/√n)

More Examples:

A sample of the reading scores of 42 fifth-graders has a mean of 78. The standard deviation of the sample is 15. Find the 95% confidence interval of the mean reading scores of all fifth-graders.   the mean reading scores of all fifth-graders is between 73 and 83
A study of 34 English composition professors showed that they spent, on average, 13 minutes correcting a student's term paper. Find the 90% confidence interval of the mean time for all composition papers when sigma = 3 minutes.  the mean reading scores of all fifth-graders is between 12.2 and 13.8

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