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Section 7.4 Estimation of a Population Mean (s is unknown )

Section 7.4 Estimation of a Population Mean (s is unknown ). This section presents methods for estimating a population mean when the population standard deviation s is not known. Best Point Estimate. The sample mean x is still the best point estimate of the population mean m. _.

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Section 7.4 Estimation of a Population Mean (s is unknown )

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  1. Section 7.4Estimation of a Population Mean(s is unknown) This section presents methods for estimating apopulation mean when the population standard deviation sis not known.

  2. Best Point Estimate The sample mean x is still the best point estimate of the population meanm. _

  3. Student t Distribution ( t-dist ) When σis unknown, we must use the Student t distribution instead of the normal distribution. Requires new parameterdf = Degrees of Freedom

  4. Definition • The number of degrees of freedom (df)for a collection of sample data is defined as: • “The number of sample values that can vary after certain restrictions have been imposed on all data values.” • In this section:df = n – 1 • Basically, since σ is unknown, a data point has to be “sacrificed” to make s. So all further calculations use n – 1 data points instead of n.

  5. Using the Student t Distribution • Thet-scoreis similar to the z-score but applies for the t-dist instead of the z-dist. The same is true for probabilities and critical values. • P(t < -1) tα • (Area under curve) (Critical value) • NOTE: The values depend on df α(area) 0 -1 0

  6. Important Properties of the Student t Distribution • Has a symmetric bell shapesimilar to the z-dist • Has a wider distribution than that the z-dist • Mean μ = 0 • S.D. σ > 1 (Note: σ varies with df) • As df gets larger, the t-dist approaches the z-dist

  7. Student t Distributions for n = 3 and n = 12

  8. z-Distribution andt-Distribution df= 2 df= 100 Wider Spread Almost the same As df increases, the t-distapproaches the z-dist

  9. Progression of t-dist with df df= 2 df= 3 df= 4 df= 6 df= 7 df= 8 df= 20 df= 50 df= 100 df= 5

  10. Choosing the Appropriate Distribution known and normally distributed population or known andn > 30 Use the normal (Z) distribution not known and normally distributed population or not known andn > 30 Use tdistribution Population is not normally distributed andn ≤ 30 Methods of Ch. 7do not apply

  11. Calculating values from t-dist Stat → Calculators → T

  12. Calculating values from t-dist Enter Degrees of Freedom (DF) and t-score

  13. Calculating values from t-dist P(t<-1) = 0.1646 when df = 20

  14. Calculating values from t-dist tα = 1.697 whenα = 0.05 df = 20

  15. Formula 7-6 wheret/2hasn – 1degrees of freedom. Margin of Error E for Estimate of  (σunknown) t/2=The t-value separating the right tail so it has an area of/2

  16. C.I. for the Estimate of μ(With σ Not Known)

  17. Finding the Point Estimate and E from a C.I. Point estimate of µ: Margin of Error:

  18. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s Note: Same parameters as example used in Section 7-37-3: Etimating a population mean:σ known Using σ = 10 ( instead of s = 10.0 ) we found the 90% confidence interval: C.I. = (35.9, 40.9)

  19. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Direct Computation: T Calculator (df= 41)

  20. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Using StatCrunch Stat → T statistics → One Sample → with Summary

  21. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Using StatCrunch Enter Parameters, click Next

  22. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Using StatCrunch Select Confidence Interval and enter Confidence Level, then click Calculate

  23. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s .0 Using StatCrunch Standard Error Lower Limit Upper Limit From the output, we find the Confidence interval is CI = (35.8, 41.0)

  24. Example: Find the 90%confidence interval for the population mean using a sample of size 42, mean 38.4, and standard deviation 10.0 s Results If σknown Used σ = 10 to obtain 90% CI: (35.9, 40.9) If σunknown Used s = 10.0 to obtain 90% CI: (35.8, 41.0) Notice: σ known yields a smaller CI (i.e. less uncertainty)

  25. Section 7.5Estimation of a Population Variance This section presents methods for estimating apopulation variances2and standard deviation s.

  26. Best Point Estimate of s2 The sample variance s2is the best point estimate of the population variances2

  27. Best Point Estimate of s Thesample standard deviation s is the best point estimateof the population standard deviations

  28. The 2 Distribution ( 2-dist ) Pronounced “Chi-squared” Also dependent on the number degrees of freedom df.

  29. Properties of the 2 Distribution The chi-square distribution is not symmetric, unlike the z-dist and t-dist. The values can be zero or positive, they are nonnegative. Dependent on the Degrees of Freedom: df = n – 1 Chi-Square Distribution for df = 10 and df = 20 Chi-Square Distribution Use StatCrunch to Calculate values (similar to z-dist and t-dist)

  30. Calculating values from 2-dist Stat → Calculators → Chi-Squared

  31. Calculating values from 2-dist Enter Degrees of Freedom DF and parameters( same procedure as with t-dist ) P(2 < 10)= 0.5595 when df = 10

  32. Example: Find the 90% left and rightcritical values (2Land 2R) of the 2-dist when df= 20 Need to calculate values when the left/right areas are 0.05 ( i.e. α/2 ) 2L= 10.851 2R= 31.410

  33. Important Note!! The 2-distribution is used for calculating the Confidence Interval of the Variance σ2 Take the square-root of the values to get the Confidence Interval of the Standard Deviation σ ( This is why we call it 2 instead of )

  34. Confidence Interval for Estimating a Population Variance Note: Left and Right Critical values on opposite sides

  35. Confidence Interval for Estimating a Population Standard Deviation Note: Left and Right Critical values on opposite sides

  36. Requirement for Application The population MUST be normally distributed to hold(even when using large samples) This requirement is very strict!

  37. Round-Off Rules for Confidence Intervals Used to Estimate  or  2 • When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data. • When the original set of data is unknown and only the summary statistics(n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample standard deviation.

  38. Example Suppose the scores a test follow a normal distribution. Given a sample of size 40with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation. Direct Computation: Chi-Squared Calculator (df= 39)

  39. Example Suppose the scores a test follow a normal distribution. Given a sample of size 40with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation. Using StatCrunch Stat → Variance → One Sample → with Summary

  40. Example Suppose the scores a test follow a normal distribution. Given a sample of size 40with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation. Using StatCrunch Sample Variance Enter parameters, then click Next Be sure to enter the sample variance s2 (not s)

  41. Example Suppose the scores a test follow a normal distribution. Given a sample of size 40with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation. Using StatCrunch Select Confidence Interval, enter Confidence Level, then click Calculate

  42. Example Suppose the scores a test follow a normal distribution. Given a sample of size 40with mean 72.8 and standard deviation 4.92, find the 95% C.I. of the population standard deviation. Using StatCrunch Remember: The result is the C.I for the Variance σ2 Take the square root for Standard Deviation σ Variance Lower Limit: LLσ2 Variance Upper Limit: ULσ2 σ2 CI = (LLσ2, ULσ2) = (16.2, 39.9) σ CI = (LLσ2, ULσ2) = (4.03, 6.32)

  43. Determining Sample Sizes The procedure for finding the sample size necessary to estimate 2 is based on Table 7-2 You just read the required sample size from an appropriate line of the table.

  44. Table 7-2

  45. Example We want to estimate the standard deviation . We want to be 95% confident that our estimate is within 20%of the true value of . Assume that the population is normally distributed. How large should the sample be? For 95% confident andwithin 20% From Table 7-2 (see next slide), we can see that 95% confidence and an error of 20% for  correspond to a sample of size 48. We should obtain a sample of 48 values.

  46. For 95% confident andwithin 20%

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