1. Population All members of a set which have a given characteristic. Population Data Data associated with a certain population. Population Parameter A measure that is computed using population data. Sample A subset of the population. 7.1 Definitions

2. Statistic An estimated population parameter computed from the data in the sample. Random Sample A sample of which every member of the population has an equal chance of being a part. Bias A study that favors certain outcomes. Point Estimate A single number (point) that attempts to estimate an unknown parameter.

3. Population (Size of population = N) Sample number 1 Sample number 2 Sample number 3 Sample number N C n Each sample size = n

4. Example: Let N = {2,6,10,11,21} Find µ, median and σ µ = 10 median = 10 σ = 6.36 How many samples of size 3 are possible? navemedSxSx σ 1 2,6,1066 43.26 2 2,6,116.336 4.53.68 3 2,6,219.676 10.018.17 4 2,10,117.6710 4.934.03 5 2,10,211110 9.537.79 6 2,11,2111.3311 9.507.76 7 6,10,11910 2.642.16 8 6,10,2112.3310 7.776.34 9 6,11,2112.6711 7.636.24 10 10,11,211411 6.084.97 5 C 3 = 10 med 6 10 11 P(x).3.4.3

5. Point Estimators Sample Population

6. 1 092725 012157 827052 297980 625608 964134 2 104460 007903 484595 868313 274221 367181 3 676071 388003 266711 323324 044463 762803 4 881878 862385 203886 261061 096674 811548 5 534500 336348 086585 241740 581286 008435 6 094276 615776 242112 985859 075388 082003 7 333848 513630 474798 841425 331001 542740 8 847886 629263 596457 589243 576797 800957 9 942495 695172 523982 264961 771016 118797 10 450553 679145 324036 715835 963418 533048 11 024670 615375 717260 171144 340939 208712 12 932959 205554 113225 704406 263818 633643 Random Number Table

7. Characteristics of a Sampling Distribution Definition: The sampling distribution of the X’s is a frequency curve, or histogram constructed from all the N C n possible values of X. Characteristic 2. The standard deviation of the sampling distribution which is the standard deviation of all the N C n of X is equal to the standard deviation of the population divided by the square root of the sample size. (Also called the standard error, SE.) Characteristic 1. The mean of all the N C n possible values of X is equal to the population mean, µ.

8. Assumptions  n > 30 The sample must have more than 30 values.  Simple Random Sample All samples of the same size have an equal chance of being selected. Large Samples

9. Definitions  Estimator a formula or process for using sample data to estimate a population parameter  Estimate a specific value or range of values used to approximate some population parameter  Point Estimate a single value (or point) used to approximate a population parameter

10. The sample mean x is the best point estimate of the population mean µ. The sample standard deviation s is the best point estimate of the population standard deviation . The sample proportion p is the best point estimate of the population proportion . Definitions

11. Confidence Interval (or Interval Estimate) A C% confidence interval for a population mean, μ, is an interval [a,b] such that μ would lie within C% of such intervals if repeated samples of the same size were formed and interval estimates made. Central Limit Theorem Under certain conditions, the sampling distribution of the X’s result in a normal distribution

12. Definition Confidence Interval (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter Lower # < population parameter < Upper #

13. Confidence Interval (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter Lower # < population parameter < Upper # As an example Lower # <  < Upper #

14. the probability 1 -  (often expressed as the equivalent percentage value) that is the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times usually 90%, 95%, or 99% (  = 10%), (  = 5%), (  = 1%) Degree of Confidence (level of confidence or confidence coefficient)

15. Interpreting a Confidence Interval Correct: We are 95% confident that the interval from 98.08 to 98.32 actually does contain the true value of . This means that if we were to select many different samples of size 106 and construct the confidence intervals, 95% of them would actually contain the value of the population mean . Wrong: There is a 95% chance that the true value of  will fall between 98.08 and 98.32. 98.08 o < µ < 98.32 o

16. Confidence Intervals from 20 Different Samples

17. the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z  /2 is a critical value that is a z score with the property that it separates an area  / 2 in the right tail of the standard normal distribution. Critical Value

18. The Critical Value z=0 Found from Table A-2 (corresponds to area of 0.5 -   2 ) z  2 -z  2  2

19. Finding z  2 for 95% Degree of Confidence

20. -z  2 z  2 95%.95.025  2 = 2.5% =.025  = 5% Finding z  2 for 95% Degree of Confidence

21. -z  2 z  2 95%.95.025  2 = 2.5% =.025  = 5% Critical Values Finding z  2 for 95% Degree of Confidence

23. .4750.025 Use Table A-2 to find a z score of 1.96  = 0.025  = 0.05 Finding z  2 for 95% Degree of Confidence

24. .025 - 1.96 1.96 z  2 = 1.96  .4750.025 Use Table A-2 to find a z score of 1.96  = 0.025  = 0.05

25. Margin of Error Definition

26. Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E Definition

27. Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E Definition

28. Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E x -E < µ < x +E Definition

29. Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E x -E < µ < x +E lower limit Definition upper limit

30. Definition Margin of Error µ x + E x - E E = z  /2  n

31. Definition Margin of Error µ x + E x - E also called the maximum error of the estimate E = z  /2  n

32. Calculating E When  Is Unknown  If n > 30, we can replace  in Formula 6-1 by the sample standard deviation s.  If n  30, the population must have a normal distribution and we must know  to use Formula 6-1.

33. x - E < µ < x + E (x + E, x - E) µ = x + E Confidence Interval (or Interval Estimate) for Population Mean µ (Based on Large Samples: n >30)

34. Procedure for Constructing a Confidence Interval for µ ( Based on a Large Sample: n > 30 ) 1. Find the critical value z  2 that corresponds to the desired degree of confidence. 3. Find the values of x - E and x + E. Substitute those values in the general format of the confidence interval: 4. Round using the confidence intervals roundoff rules. x - E < µ < x + E 2. Evaluate the margin of error E = z  2  / n. If the population standard deviation  is unknown, use the value of the sample standard deviation s provided that n > 30.

35. Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

36. n = 106 x = 98.20 o s = 0.62 o  = 0.05  / 2 = 0.025 z  / 2 = 1.96 n E = z  / 2  = 1.96 0.62 = 0.12 106 Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. x - E <  < x + E 98.20 o - 0.12 <  < 98.20 o + 0.12 98.08 o <  < 98.32 o

37. Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval. Based on the sample provided, the confidence interval for the population mean is 98.08 o <  < 98.32 o. If we were to select many different samples of the same size, 95% of the confidence intervals would actually contain the population mean .

38. Finding the Point Estimate and E from a Confidence Interval Point estimate of µ : x = (upper confidence interval limit) + (lower confidence interval limit) 2 Margin of Error: E = (upper confidence interval limit) - (lower confidence interval limit) 2