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Topic 7 Estimation

Mathematics & Statistics Statistics. Topic 7 Estimation. Topic Goals. After completing this topic, you should be able to: Distinguish between a point estimate and a confidence interval estimate

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Topic 7 Estimation

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  1. Mathematics & Statistics Statistics Topic 7 Estimation

  2. Topic Goals After completing this topic, you should be able to: • Distinguish between a point estimate and a confidence interval estimate • Construct and interpret a confidence interval estimate for a single population mean using both the Z and t distributions • Form and interpret a confidence interval estimate for a single population proportion • Form and interpret a confidence interval estimate for a single population variance

  3. Estimation • Last week we looked at the distribution of sample statistics... • ...given the value of a population parameter. • In real-life situations we don’t know the true value of the population parameter... • ...so the question is: • Can we say something about the value of the population parameter... • ...given an observed value of a sample statistic?

  4. Confidence Intervals Content of this topic • Confidence Intervals for the Population Mean, μ • when Population Variance σ2 is Known (Section 8.2) • when Population Variance σ2 is Unknown (Section 8.3) • Confidence Intervals for the Population Proportion, p (large samples) (Section 8.4) • Confidence Intervals for the Population Variance, σ2(Section 9.4)

  5. Definitions • An estimator of a population parameteris • a random variable that depends on sample information . . . • whose value provides an approximation to this unknown parameter • A specific value of that random variable is called an estimate

  6. Point Estimates We can estimate a Population Parameter … with a SampleStatistic (a Point Estimate) μ x Mean Proportion p Variance s2 σ2 Variance

  7. Unbiasedness • A point estimator is said to be an unbiased estimator of the parameter  if the expected value, or mean, of the sampling distribution of is , • Examples: • The sample mean is an unbiased estimator of μ • The sample variance is an unbiased estimator of σ2 • The sample proportion is an unbiased estimator of p

  8. Unbiasedness (continued) • is an unbiased estimator, is biased:

  9. Most Efficient Estimator • Suppose there are several unbiased estimators of  • The most efficient estimator or the minimum variance unbiased estimator of  is the unbiased estimator with the smallest variance • Let and be two unbiased estimators of , based on the same number of sample observations. Then, • is said to be more efficient than if • The relative efficiency of with respect to is the ratio of their variances:

  10. Point and Interval Estimates • A point estimate is a single number, • a confidence interval provides additional information about variability Upper Confidence Limit Lower Confidence Limit Point Estimate Width of confidence interval

  11. Confidence Intervals • How much uncertainty is associated with a point estimate of a population parameter? • An interval estimate provides more information about a population characteristic than does a point estimate • Such interval estimates are called confidence intervals

  12. Confidence Interval Estimate • An interval gives a range of values: • Takes into consideration variation in sample statistics from sample to sample • Based on observation from 1 sample • Gives information about closeness to unknown population parameters • Stated in terms of level of confidence • Can never be 100% confident

  13. Confidence Interval and Confidence Level • If P(a <  < b) = 1 -  then the interval from a to b is called a 100(1 - )% confidence interval of . • The quantity (1 - ) is called the confidence level of the interval ( between 0 and 1) • In repeated samples of the population, the true value of the parameter  would be contained in 100(1 - )% of intervals calculated this way. • The confidence interval calculated in this manner is written as a <  < b with 100(1 - )% confidence

  14. I am 95% confident that μ is between 40 & 60. Estimation Process Random Sample Population Mean X = 50 (mean, μ, is unknown) Sample

  15. Confidence Level, (1-) (continued) • Suppose confidence level = 95% • Also written (1 - ) = 0.95 • A relative frequency interpretation: • From repeated samples, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter • A specific interval either will contain or will not contain the true parameter • The procedure used leads to a correct interval in 95% of the time... • ...but this does not guarantee anything about a particular sample.

  16. General Formula • The general formula for all confidence intervals is: • The value of the reliability factor depends on the desired level of confidence Point Estimate ± (Reliability Factor)(Standard deviation)

  17. Confidence Intervals Confidence Intervals Population Variance Population Mean Population Proportion σ2Unknown σ2Known

  18. Confidence Interval for μ(σ2 Known) • Assumptions • Population variance σ2is known • Population is normally distributed... • ....or large sample so that CLT can be used. • Confidence interval estimate: (where z/2 is the normal distribution value for a probability of /2 in each tail)

  19. Example • A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms. • Determine a 95% confidence interval for the true mean resistance of the population.

  20. Example (continued) • A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms. • Solution:

  21. Interpretation • We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms • Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

  22. Margin of Error • The confidence interval, • Can also be written as where ME is called the margin of error • The interval width, w, is equal to twice the margin of error

  23. Finding the Reliability Factor, z/2 • Consider a 95% confidence interval: z = -1.96 z = 1.96 Z units: 0 Lower Confidence Limit Upper Confidence Limit X units: Point Estimate • Find z.025 = 1.96 from the standard normal distribution table

  24. Common Levels of Confidence • Commonly used confidence levels are 90%, 95%, and 99% Confidence Coefficient, Confidence Level Z/2 value 80% 90% 95% 98% 99% 99.8% 99.9% .80 .90 .95 .98 .99 .998 .999 1.28 1.645 1.96 2.33 2.58 3.08 3.27

  25. Intervals and Level of Confidence Sampling Distribution of the Mean x Intervals extend from to x1 100(1-)%of intervals constructed contain μ; 100()% do not. x2 Confidence Intervals

  26. Summary:Finding a confidence interval for μ (σ known) • Choose confidence level 1-α (e.g. .95). • Find an interval [a,b] such that P(a<μ<b)=1-α. • a and b are determined by • How to find zα/2? • Look in table for value such that P(Z> zα/2)=α • e.g. if 1-α=.95, then zα/2 = 1.96.

  27. Finding the Reliability Factor, z/2 • Consider a 95% confidence interval: z = -1.96 z = 1.96 Z units: 0 Lower Confidence Limit Upper Confidence Limit X units: Point Estimate • Find z.025 = 1.96 from the standard normal distribution table

  28. Example • Assume that the calorie contents per 100 ml of Guiness is normally distributed. • A sample of 11 pints has a mean calorie content per 100 ml of 35.1. We know from past testing that the population standard deviation is 2.35. • Determine a 95% confidence interval for the true mean calorie content per 100 ml Guiness.

  29. Example (continued) • A sample of 11 pints from a large normal population has a mean calorie content per 100 ml of 35.1. We know from past testing that the population standard deviation is 2.35 calories per 100 ml. • Solution:

  30. Interpretation • We are 95% confident that the true calorie content per 100 ml is between 33.7112 and 36.4888. • Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

  31. Example • Suppose a second sample of 11 pints has a mean calorie content per 100 ml of 35.9. • A 95% confidence interval for this sample is:

  32. Reducing the Margin of Error The margin of error can be reduced if • the population standard deviation can be reduced (σ↓) • The sample size is increased (n↑) • The confidence level is decreased, (1 – ) ↓

  33. How is the formula obtained? • Recall the formula for the confidence interval: • How is it obtained? • A 100(1-α)% confidence interval is an interval [a,b] such that P(a<μ<b) = 1-α. • We use the fact that

  34. Derivation

  35. Large Samples • If the population is not normal.... • ....and the variance is not known.... • The same confidence interval can still be used... • ...if the sample is large. • For in that case the Central Limit Theorem tells us that the sample mean is approximately normal with mean μ and standard deviation σ/√n,... • ...and s2 ≈ σ2.

  36. Example • For a sample of 200 tea boxes you observe that the average weight is 101.0 grams with a standard deviation of 2.78 grams. • Determine a 99% confidence interval for the population mean. • Solution. Note that the sample is large, so: • σ2≈ s2 • the Central Limit Theorem says that the sample mean is approximately normal with mean μ and standard deviation σ/√n≈2.78/√200=.197

  37. Example • So, we can proceed as if the sample mean is normal with known variance and apply • Therefore:

  38. Confidence Intervals Confidence Intervals Population Variance Population Mean Population Proportion σ2Unknown σ2Known

  39. Confidence Interval for μ(σ2 Unknown) • If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s • This introduces extra uncertainty, since s is variable from sample to sample • Therefore we use the t distribution instead of the normal distribution

  40. Student’s t Distribution • Consider a random sample of n observations • with mean x and standard deviation s • from a normally distributed population with mean μ • Then the variable follows the Student’s t distribution with (n - 1) degrees of freedom

  41. Student’s t Distribution • The t is a family of distributions • The t-value depends on degrees of freedom (d.f.) • Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1

  42. Student’s t Distribution Note: t Z as n increases Standard Normal (t with df = ∞) t (df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal t (df = 5) t 0

  43. Confidence Interval for μ(σ Unknown) (continued) • Assumptions • Population standard deviation is unknown • Population is normally distributed • Use Student’s t Distribution • Confidence Interval Estimate: where tn-1,α/2 is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail:

  44. Student’s t Table Upper Tail Area Let: n = 3 df = n - 1 = 2  = .10/2 =.05 df .10 .025 .05 1 12.706 3.078 6.314 2 1.886 4.303 2.920 /2 = .05 3 3.182 1.638 2.353 The body of the table contains t values, not probabilities 0 t 2.920

  45. t distribution values With comparison to the Z value Confidence t t t Z Level (10 d.f.)(20 d.f.)(30 d.f.) ____ .80 1.372 1.325 1.310 1.282 .90 1.812 1.725 1.697 1.645 .95 2.228 2.086 2.042 1.960 .99 3.169 2.845 2.750 2.576 Note: t Z as n increases

  46. Example • A sample of 11 pints has a mean calorie content per 100 ml of 35.1, with a standard deviation of 2.35. • Determine a 95% confidence interval for the true mean calorie content per 100 ml Guiness if the calorie content is normal.

  47. Example (continued) • A sample of 11 pints from a large normal population has a mean calorie content per 100 ml of 35.1, with a standard deviation of 2.35 calories per 100 ml. • Solution:

  48. Interpretation • We are 95% confident that the true calorie content per 100 ml is between 33.5213 and 36.6787. • Compare with the previous example (variance known)... • ...where we were 95% confident that the true calorie content per 100 ml is between 33.7112 and 36.4888. • The interval is wider because there is additional uncertainty over σ.

  49. About Student • Student is the pseudonym of William Sealy Gosset... • ...who worked for the Guiness brewery in Dublin... • ...and needed procedures for quality control of the brewing process. • He derived the distribution of which is still named after him

  50. Confidence Intervals Confidence Intervals Population Variance Population Mean Population Proportion σ2Unknown σ2Known

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