Confidence interval

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Name                                       Shakeel Nouman
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Domicile                            Punjab (Lahore)
Contact #             0332-4462527. 0321-9898767
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Confidence interval

  1. 1. Confidence Intervals Slide 1 Shakeel Nouman M.Phil Statistics Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  2. 2. 6        Slide 2 Confidence Intervals Using Statistics Confidence Interval for the Population Mean When the Population Standard Deviation is Known Confidence Intervals for  When  is Unknown - The t Distribution Large-Sample Confidence Intervals for the Population Proportion p Confidence Intervals for the Population Variance Sample Size Determination Summary and Review of Terms Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  3. 3. 6-1 Introduction • Slide 3 Consider the following statements: x = 550 • A single-valued estimate that conveys little information about the actual value of the population mean. We are 99% confident that is in the interval [449,551] • An interval estimate which locates the population mean within a narrow interval, with a high level of confidence. We are 90% confident that is in the interval [400,700] • An interval estimate which locates the population mean within a broader interval, with a lower level of confidence. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  4. 4. Types of Estimators • Slide 4 Point Estimate A single-valued estimate. A single element chosen from a sampling distribution. Conveys little information about the actual value of the population parameter, about the accuracy of the estimate. • Confidence Interval or Interval Estimate An interval or range of values believed to include the unknown population parameter. Associated with the interval is a measure of the confidence we have that the interval does indeed contain the parameter of interest. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  5. 5. Confidence Interval or Interval Estimate Slide 5 A confidence interval or interval estimate is a range or interval of numbers believed to include an unknown population parameter. Associated with the interval is a measure of the confidence we have that the interval does indeed contain the parameter of interest. • A confidence interval or interval estimate has two components: A range or interval of values An associated level of confidence Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  6. 6. 6-2 Confidence Interval for  When  Is Known • If the population distribution is normal, the sampling distribution of the mean is normal. If the sample is sufficiently large, regardless of the shape of the population distribution, the sampling distribution is normal (Central Limit Theorem). In either case: Standard Normal Distribution: 95% Interval 0.4     P   196 .  x    196   0.95 .  n n 0.3 f(z)  Slide 6 or 0.2 0.1   P x  196 .  n      x  196   0.95 . n 0.0 -4 -3 -2 -1 0 1 2 3 4 z Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  7. 7. 6-2 Confidence Interval for  when  is Known (Continued) Slide 7 Before sampling, there is a 0.95probability that the interval   1.96  n will include the sample mean (and 5% that it will not). Conversely, after sampling, approximately 95% of such intervals x  1.96  n will include the population mean (and 5% of them will not). That is, x  1.96  n is a 95% confidenceinterval for  . Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  8. 8. A 95% Interval around the Population Mean Slide 8 Sampling Distribution of the Mean Approximately 95% of sample means can be expected to fall within the interval    196  ,   196   . . .  n n 0.4 95% f(x) 0.3 0.2 0.1 2.5%  2.5% 0.0   1.96     196 . n  x n x x 2.5% fall below the interval x  Conversely, about 2.5% can be  expected to be above   1.96 n and 2.5% can be expected to be below    1.96 . n x x 2.5% fall above the interval x x x x 95% fall within the interval So 5% can be expected to fall outside the interval       1.96 ,   1.96 .  n n   Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  9. 9. 95% Intervals around the Sample Mean Sampling Distribution of the Mean Approximately 95% of the intervals  around the sample mean can be x  1.96 n expected to include the actual value of the population mean,  (When the sample . mean falls within the 95% interval around the population mean.) 0.4 95% f(x) 0.3 0.2 0.1 2.5% 2.5% 0.0   1.96     196 . n x  x x x *5% of such intervals around the sample n mean can be expected not to include the actual value of the population mean. (When the sample mean falls outside the 95% interval around the population mean.) x x * Slide 9 x x x x x x x x * x Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  10. 10. The 95% Confidence Interval for  Slide 10 A 95% confidence interval for when is known and sampling is done from a normal population, or a large sample  is used: x  1.96 n The quantity1.96  n the sampling error. is often called the margin of error or For example, if: n = 25 A 95% confidence interval:  20 = 20 x  1.96  122  1.96 n 25 x = 122  122  (1.96)(4 )  122  7.84  114.16,129.84 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  11. 11. A (1- )100% Confidence Interval for  Slide 11 We define z as the z value that cuts off a right-tail area of  under the standard 2 2 normal curve. (1- is called the confidence coefficient. is called the error ) probability, and (1- )100% is called the confidence level. æ ö P çz > za = a/2 è ø 2 æ ö P çz < - za = a/2 è ø 2 æ ö - za < z < za = (1 - a ) Pç è 2 ø 2 S t a n d ard N o r m al Dis trib utio n 0.4 (1   ) f(z) 0.3 0.2 0.1   2 2 0.0 -5 -4 -3 -2 -1 z  2 0 1 2 Z z 2 3 4 5 (1- a)100% Confidence Interval: s x za n 2 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  12. 12. Critical Values of z and Levels of Confidence 0.99 0.98 0.95 0.90 0.80 2 0.005 0.010 0.025 0.050 0.100 Stand ard N o rm al Distrib utio n z 0.4 (1   ) 2 2.576 2.326 1.960 1.645 1.282 0.3 f(z) (1   )  Slide 12 0.2 0.1   2 2 0.0 -5 -4 -3 -2 -1 z  2 0 1 2 Z 3 z 2 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer 4 5
  13. 13. The Level of Confidence and the Width of the Confidence Interval Slide 13 When sampling from the same population, using a fixed sample size, the higher the confidence level, the wider the confidence S t a n d a r d N o r m al Di s tri b u ti o n interval. S t a n d ar d N o r m al Di s tri b uti o n 0 .3 f(z) 0 .4 0.3 f(z) 0.4 0.2 0.1 0 .2 0 .1 0.0 0 .0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 Z 80% Confidence Interval: x  128 . 0 1 2 3 4 Z  n 95% Confidence Interval: x  196 .  n Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer 5
  14. 14. The Sample Size and the Width of the Confidence Interval Slide 14 When sampling from the same population, using a fixed confidence level, the larger the sample size, n, the narrower the confidence S S a m p lin g D is trib u tio n o f th e Me a n interval.a m p lin g D is trib u tio n o f th e Me a n 0 .4 0 .9 0 .8 0 .7 0 .3 f(x) f(x) 0 .6 0 .2 0 .5 0 .4 0 .3 0 .1 0 .2 0 .1 0 .0 0 .0 x x 95% Confidence Interval: n = 20 95% Confidence Interval: n = 40 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  15. 15. Example 6-1 Slide 15 • Population consists of the Fortune 500 Companies (Fortune Web Site), as ranked by Revenues. You are trying to to find out the average Revenues for the companies on the list. The population standard deviation is $15,056.37. A random sample of 30 companies obtains a sample mean of $10,672.87. Give a 95% and 90% confidence interval for the average Revenues. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  16. 16. Example 6-1 (continued) - Using the Template Slide 16 Note: The remaining part of the template display is shown on the next slide. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  17. 17. Example 6-1 (continued) - Using the Template Slide 17  (Sig) Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  18. 18. Slide Example 6-1 (continued) - Using the 18 Template when the Sample Data is Known Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  19. 19. 6-3 Confidence Interval or Interval Estimate for  When  Is Unknown The t Distribution Slide 19 If the population standard deviation,  is not known, replace ,  with the sample standard deviation, s. If the population is normal, X  the resulting statistic: t  has a t distribution with (n - 1) • • • • s n degrees The t is a family of bell-shaped and symmetric distributions, one for each number of degree of freedom. The expected value of t is 0. For df > 2, the variance of t is df/(df-2). This is greater than 1, but approaches 1 as the number of degrees of freedom increases. The t is flatter and has fatter tails than does the standard normal. The t distribution approaches a standard normal as the number of degrees of freedom increases of freedom. Standard normal t, df = 20 t, df = 10   Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  20. 20. The t Distribution Template Slide 20 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  21. 21. 6-3 Confidence Intervals for  when  is Unknown- The t Distribution Slide 21 A (1- )100% confidence interval for when is not known (assuming a normally distributed population): x t  2 s n where t is the value of the t distribution with n-1 degrees of 2  2 freedom that cuts off a tail area of to its right. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  22. 22. The t Distribution t D is trib u tio n : d f= 1 0 -----0 .4 0 .3 Area = 0.10 0 .2 Area = 0.10 } t0.005 -----63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.358 2.326 } t0.010 -----31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 1.980 1.960 f(t) t0.025 ----12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.658 1.645 0 .1 0 .0 -2.228 Area = 0.025 2.617 2.576 -1.372 0 t 1.372 2.228 } t0.050 ----6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.289 1.282 } df t0.100 --13.078 21.886 31.638 41.533 51.476 61.440 71.415 81.397 91.383 101.372 111.363 121.356 131.350 141.345 151.341 161.337 171.333 181.330 191.328 201.325 211.323 221.321 231.319 241.318 251.316 261.315 271.314 281.313 291.311 301.310 401.303 601.296 120 Slide 22 Area = 0.025 Whenever is not known (and the population is assumed normal), the correct distribution to use is the t distribution with n-1 degrees of freedom. Note, however, that for large degrees of freedom, the t distribution is approximated well by the Z distribution.  Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  23. 23. Example 6-2 Slide 23 A stock market analyst wants to estimate the average return on a certain stock. A random sample of 15 days yields an average (annualized) return of x  10.37% and a standard deviation of s = 3.5%. Assuming a normal population of returns, give a 95% confidence interval for the average return on this stock. df --1 . . . 13 14 15 . . . t0.100 ----3.078 . . . 1.350 1.345 1.341 . . . t0.050 ----6.314 . . . 1.771 1.761 1.753 . . . t0.025 -----12.706 . . . 2.160 2.145 2.131 . . . t0.010 -----31.821 . . . 2.650 2.624 2.602 . . . t0.005 -----63.657 . . . 3.012 2.977 2.947 . . . The critical value of t for df = (n -1) = (15 -1) =14 and a right-tail area of 0.025 is: t 0.025  2.145 The corresponding confidence interval or interval estimate is: s x  t 0.025 n 35 .  10.37  2.145 15  10.37  1.94  8.43,12.31 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  24. 24. Large Sample Confidence Intervals for the Population Mean df t0.100 --13.078 .. .. .. 120  t0.050 ----6.314 . . . 1.289 1.282 t0.025 ----12.706 . . . 1.658 1.645 t0.010 -----31.821 . . . 1.980 1.960 t0.005 -----63.657 . . . 2.358 2.326 ------ 2.617 2.576 Slide 24 Whenever is not known (and the population is assumed normal), the correct distribution to use is the t distribution with n-1 degrees of freedom. Note, however, that for large degrees of freedom, the t distribution is approximated well by the Z distribution. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  25. 25. Large Sample Confidence Intervals for the Population Mean Slide 25 A large - sample (1-  )100% confidence interval for : s x  z n 2 Example 6-3: An economist wants to estimate the average amount in checking accounts at banks in a given region. A random sample of 100 accounts gives x-bar = $357.60 and s = $140.00. Give a 95% confidence interval for  the , average amount in any checking account at a bank in the given region. x  z 0.025 s 140.00  357.60  1.96  357.60  27.44   33016,385.04 . n 100 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  26. 26. 6-4 Large-Sample Confidence Intervals for the Population Proportion, p Slide 26 The estimator of the population proportion, p , is the sample proportion, p. If the  sample size is large, p has an approximately normal distribution, with E( p ) = p and   pq V( p ) = , where q = (1 - p). When the population proportion is unknown, use the  n estimated value, p , to estimate the standard deviation of p .   For estimating p , a sample is considered large enough when both n  p an n  q are greater than 5. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  27. 27. 6-4 Large-Sample Confidence Intervals for the Population Proportion, p Slide 27 A large - sample (1 -  )100% confidence interval for the population proportion, p : pz ˆ α 2 pq ˆˆ n ˆ where the sample proportion, p, is equal to the number of successes in the sample, x, ˆ ˆ divided by the number of trials (the sample size), n, and q = 1 - p. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  28. 28. Large-Sample Confidence Interval for the Population Proportion, p (Example 6-4) Slide 28 A marketing research firm wants to estimate the share that foreign companies have in the American market for certain products. A random sample of 100 consumers is obtained, and it is found that 34 people in the sample are users of foreign-made products; the rest are users of domestic products. Give a 95% confidence interval for the share of foreign products in this market. p  z  2 pq ( 0.34 )( 0.66)   0.34  1.96 n 100  0.34  (1.96)( 0.04737 )  0.34  0.0928   0.2472 ,0.4328 Thus, the firm may be 95% confident that foreign manufacturers control anywhere from 24.72% to 43.28% of the market. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  29. 29. Slide 29 Large-Sample Confidence Interval for the Population Proportion, p (Example 6-4) – Using the Template Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  30. 30. Reducing the Width of Confidence Intervals - The Value of Information Slide 30 The width of a confidence interval can be reduced only at the price of: • a lower level of confidence, or • a larger sample. Lower Level of Confidence Larger Sample Size Sample Size, n = 200 90% Confidence Interval p  z  2 pq  (0.34)(0.66)  0.34  1645 . n 100  0.34  (1645)(0.04737) .  0.34  0.07792  0.2621,0.4197 p  z  2 pq  (0.34)(0.66)  0.34  196 . n 200  0.34  (196)(0.03350) .  0.34  0.0657  0.2743,0.4057 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  31. 31. 6-5 Confidence Intervals for the 2 Population Variance: The Chi-Square ( ) Distribution Slide 31 • The sample variance, s2, is an unbiased • 2 estimator of the population variance,  . Confidence intervals for the population 2 variance are based on the chi-square ( ) distribution. The chi-square distribution is the probability distribution of the sum of several independent, squared standard normal random variables. The mean of the chi-square distribution is equal to 2 the degrees of freedom parameter, (E[ ] = df). The variance of a chi-square is equal to twice the number 2 of degrees of freedom, (V[ ] = 2df). Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  32. 32. The Chi-Square (2) Distribution  C hi-S q uare D is trib utio n: d f=1 0 , d f=3 0 , d f =5 0 0 .1 0 df = 10 0 .0 9 0 .0 8 0 .0 7 0 .0 6 2  The chi-square random variable cannot be negative, so it is bound by zero on the left. The chi-square distribution is skewed to the right. The chi-square distribution approaches a normal as the degrees of freedom increase. f( )  Slide 32 df = 30 0 .0 5 0 .0 4 df = 50 0 .0 3 0 .0 2 0 .0 1 0 .0 0 0 50 100 2 In sampling from a normal population, the random variable:   2 ( n  1) s 2 2 has a chi - square distribution with (n - 1) degrees of freedom. Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  33. 33. Values and Probabilities of ChiSquare Distributions Slide 33 Area in Right Tail .995 .990 .975 .950 .900 .100 .050 .025 .010 .005 .900 .950 .975 .990 .995 Area in Left Tail df 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .005 0.0000393 0.0100 0.0717 0.207 0.412 0.676 0.989 1.34 1.73 2.16 2.60 3.07 3.57 4.07 4.60 5.14 5.70 6.26 6.84 7.43 8.03 8.64 9.26 9.89 10.52 11.16 11.81 12.46 13.12 13.79 .010 .025 .050 .100 0.000157 0.0201 0.115 0.297 0.554 0.872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.20 10.86 11.52 12.20 12.88 13.56 14.26 14.95 0.000982 0.0506 0.216 0.484 0.831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.28 10.98 11.69 12.40 13.12 13.84 14.57 15.31 16.05 16.79 0.000393 0.103 0.352 0.711 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.12 10.85 11.59 12.34 13.09 13.85 14.61 15.38 16.15 16.93 17.71 18.49 0.0158 0.211 0.584 1.06 1.61 2.20 2.83 3.49 4.17 4.87 5.58 6.30 7.04 7.79 8.55 9.31 10.09 10.86 11.65 12.44 13.24 14.04 14.85 15.66 16.47 17.29 18.11 18.94 19.77 20.60 2.71 4.61 6.25 7.78 9.24 10.64 12.02 13.36 14.68 15.99 17.28 18.55 19.81 21.06 22.31 23.54 24.77 25.99 27.20 28.41 29.62 30.81 32.01 33.20 34.38 35.56 36.74 37.92 39.09 40.26 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.68 21.03 22.36 23.68 25.00 26.30 27.59 28.87 30.14 31.41 32.67 33.92 35.17 36.42 37.65 38.89 40.11 41.34 42.56 43.77 5.02 7.38 9.35 11.14 12.83 14.45 16.01 17.53 19.02 20.48 21.92 23.34 24.74 26.12 27.49 28.85 30.19 31.53 32.85 34.17 35.48 36.78 38.08 39.36 40.65 41.92 43.19 44.46 45.72 46.98 6.63 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64 46.96 48.28 49.59 50.89 7.88 10.60 12.84 14.86 16.75 18.55 20.28 21.95 23.59 25.19 26.76 28.30 29.82 31.32 32.80 34.27 35.72 37.16 38.58 40.00 41.40 42.80 44.18 45.56 46.93 48.29 49.65 50.99 52.34 53.67 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  34. 34. Values and Probabilities of ChiSquare Distributions Slide 34 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  35. 35. Confidence Interval for the Population Variance Slide 35 A (1-)100% confidence interval for the population variance * (where the population is assumed normal) is:  2 2  ( n  1) s , ( n  1) s  2   2   1   2 2 2   is the value of the chi-square distribution with n - 1 degrees of freedom where 2 that cuts off an area  to its right and 2  is the value of the distribution that  1 cuts off an area of  2to its left (equivalently, an area of 2  to its right). 1 2 2 * Note: Because the chi-square distribution is skewed, the confidence interval for the population variance is not symmetric Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  36. 36. Confidence Interval for the Population Variance - Example 6-5 Slide 36 In an automated process, a machine fills cans of coffee. If the average amount filled is different from what it should be, the machine may be adjusted to correct the mean. If the variance of the filling process is too high, however, the machine is out of control and needs to be repaired. Therefore, from time to time regular checks of the variance of the filling process are made. This is done by randomly sampling filled cans, measuring their amounts, and computing the sample variance. A random sample of 30 cans gives an estimate s2 = 18,540. 2 Give a 95% confidence interval for the population variance,  .  2 2  ( n  1) s , ( n 21) s    ( 30  1)18540 , ( 30  1)18540   11765,33604 2  457 . 16.0         1   2 2 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  37. 37. Example 6-5 (continued) Slide 37 Area in Right Tail df . . . 28 29 30 .995 . . . 12.46 13.12 13.79 .990 . . . 13.56 14.26 14.95 .975 .950 . . . 15.31 16.05 16.79 .900 . . . 16.93 17.71 18.49 . . . 18.94 19.77 20.60 .100 .050 . . . 37.92 39.09 40.26 . . . 41.34 42.56 43.77 .025 . . . 44.46 45.72 46.98 .010 . . . 48.28 49.59 50.89 .005 . . . 50.99 52.34 53.67 Chi-Square Distribution: df = 29 0.06 0.05 0.95 f(2 ) 0.04 0.03 0.02 0.025 0.025 0.01 0.00 0 10 20  2.975  16.05 0 30 40 2 50 60 70  2.025  4572 . 0 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  38. 38. Example 6-5 Using the Template Slide 38 Using Data Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  39. 39. 6-6 Sample-Size Determination Slide 39 Before determining the necessary sample size, three questions must be answered: • How close do you want your sample estimate to be to the unknown • • parameter? (What is the desired bound, B?) What do you want the desired confidence level (1- to be so that the ) distance between your estimate and the parameter is less than or equal to B? What is your estimate of the variance (or standard deviation) of the population in question? n } For example: A (1-  ) Confidence Interval for : x  z   2 Bound, B Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  40. 40. Sample Size and Standard Error Slide 40 The sample size determines the bound of a statistic, since the standard error of a statistic shrinks as the sample size increases: Sample size = 2n Standard error of statistic Sample size = n Standard error of statistic  Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  41. 41. Minimum Sample Size: Mean and Proportion Slide 41 Minimum required sample size in estimating the population mean, m: z2 s 2 a n= 2 2 B Bound of estimate: s B = za n 2 Minimum required sample size in estimating the population proportion, p $ z2 pq a n= 2 2 B Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  42. 42. Sample-Size Determination: Example 6-6 Slide 42 A marketing research firm wants to conduct a survey to estimate the average amount spent on entertainment by each person visiting a popular resort. The people who plan the survey would like to determine the average amount spent by all people visiting the resort to within $120, with 95% confidence. From past operation of the resort, an estimate of the population standard deviation is s = $400. What is the minimum required sample size? z  2 n 2 2 B 2 2 (196) (400) .  120 2 2  42.684  43 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  43. 43. Sample-Size for Proportion: Example 6-7 Slide 43 The manufacturers of a sports car want to estimate the proportion of people in a given income bracket who are interested in the model. The company wants to know the population proportion, p, to within 0.01 with 99% confidence. Current company records indicate that the proportion p may be around 0.25. What is the minimum required sample size for this survey? n 2 z pq 2 B2 2.5762 (0.25)(0.75)  0102 .  124.42  125 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  44. 44. 6-7 The Templates – Optimizing Population Mean Estimates Slide 44 Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  45. 45. Slide 6-7 The Templates – Optimizing Population 45 Proportion Estimates Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
  46. 46. Slide 46 Name Religion Domicile Contact # E.Mail M.Phil (Statistics) Shakeel Nouman Christian Punjab (Lahore) 0332-4462527. 0321-9898767 sn_gcu@yahoo.com sn_gcu@hotmail.com GC University, . (Degree awarded by GC University) M.Sc (Statistics) Statitical Officer (BS-17) (Economics & Marketing Division) GC University, . (Degree awarded by GC University) Livestock Production Research Institute Bahadurnagar (Okara), Livestock & Dairy Development Department, Govt. of Punjab Confidence Intervals By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer

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