Lecture 2 INTERVAL ESTIMATION II

Transcription

1 Lecture 2 INTERVAL ESTIMATION II

2 Recap Population of interest - want to say something about the population mean µ perhaps Take a random sample...

3 Recap When our random sample follows a normal distribution, or indeed any distribution (if the sample size is large), then the sample mean X N(µ,σ 2 /n).

4 Recap When our random sample follows a normal distribution, or indeed any distribution (if the sample size is large), then the sample mean X N(µ,σ 2 /n). Slide/squash to give Z = X µ σ 2 /n, where Z is the standard normal distribution, i.e. Z N(0,1).

5 Recap When our random sample follows a normal distribution, or indeed any distribution (if the sample size is large), then the sample mean X N(µ,σ 2 /n). Slide/squash to give Z = X µ σ 2 /n, where Z is the standard normal distribution, i.e. Z N(0,1). Consequently, Pr( 1.96 < Z < 1.96) = 0.95 so that Pr( 1.96 < X µ < 1.96) = 0.95 σ 2 /n

6 Recap Rearranging the inequality for µ gives the 95% confidence interval as ( x 1.96 σ 2 /n, x σ 2 /n) which we can write concisely as x ±1.96 σ 2 /n.

7 Recap Rearranging the inequality for µ gives the 95% confidence interval as ( x 1.96 σ 2 /n, x σ 2 /n) which we can write concisely as x ±1.96 σ 2 /n. We have assumed that the population variance σ 2 is known! What if it isn t? (It typically isn t known in practice!)

8 Recap Rearranging the inequality for µ gives the 95% confidence interval as ( x 1.96 σ 2 /n, x σ 2 /n) which we can write concisely as x ±1.96 σ 2 /n. We have assumed that the population variance σ 2 is known! What if it isn t? (It typically isn t known in practice!) Grrrr!

9 How can we proceed? We could calculate the sample variance which we denote by s 2.

10 How can we proceed? We could calculate the sample variance which we denote by s 2. We could then estimate σ 2 with s 2.

11 How can we proceed? We could calculate the sample variance which we denote by s 2. We could then estimate σ 2 with s 2. Thus, we can think about the quantity X µ S 2 /n and what distribution this quantity might follow.

12 How can we proceed? We could calculate the sample variance which we denote by s 2. We could then estimate σ 2 with s 2. Thus, we can think about the quantity X µ S 2 /n and what distribution this quantity might follow. We will call this quantity T for reasons that will become clear from the next slide!

13 Case 2: Unknown variance σ 2 If the population variance is unknown (which is usually the case), the quantity T = X µ S 2 /n does not have a N(0,1) distribution, but a Student s t distribution.

14 Case 2: Unknown variance σ 2 If the population variance is unknown (which is usually the case), the quantity T = X µ S 2 /n does not have a N(0,1) distribution, but a Student s t distribution. This is similar to the normal distribution (i.e. symmetric and bell shaped), but is more heavily tailed ;

15 Case 2: Unknown variance σ 2 If the population variance is unknown (which is usually the case), the quantity T = X µ S 2 /n does not have a N(0,1) distribution, but a Student s t distribution. This is similar to the normal distribution (i.e. symmetric and bell shaped), but is more heavily tailed ; The t distribution has one parameter, called the degrees of freedom (ν = n 1).

16 Case 2: Unknown variance σ 2 If the population variance is unknown (which is usually the case), the quantity T = X µ S 2 /n does not have a N(0,1) distribution, but a Student s t distribution. This is similar to the normal distribution (i.e. symmetric and bell shaped), but is more heavily tailed ; The t distribution has one parameter, called the degrees of freedom (ν = n 1). A picture (using the space on page 7) will help!

17 comparison of Normal and T distributions

18 comparison of Normal and T distributions

19 Student s t distribution a brief history Takes its name from William Sealy Gosset s 1908 paper in Biometrika under the pseudonym Student. Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the chemical properties of barley where sample sizes might be small. One version of the origin of the pseudonym is that Gosset s employer forbade members of its staff from publishing scientific papers, so he had to hide his identity. Another is that Guinness did not want their competitors to know that they were using the t to test the quality of raw material.

20 William Sealy Gosset

21 Back to our problem... So if we don t know σ 2, the formula for the confidence interval becomes: x ±t p/2 s 2 /n

22 Back to our problem... So if we don t know σ 2, the formula for the confidence interval becomes: x ±t p/2 s 2 /n where t p/2 is the value such that Pr( t p/2 < T < t p/2 ) = 100(1 p)%. We find t p/2 from statistical tables (table 1.1 in the notes). We read along the p column and down the ν row.

23 Back to our problem... So if we don t know σ 2, the formula for the confidence interval becomes: x ±t p/2 s 2 /n where t p/2 is the value such that Pr( t p/2 < T < t p/2 ) = 100(1 p)%. We find t p/2 from statistical tables (table 1.1 in the notes). We read along the p column and down the ν row. For a 90% confidence interval, p = 10%. For a 95% confidence interval, p = 5%. For a 99% confidence interval, p = 1%. The degrees of freedom, ν = n 1.

24 Example (page 8) A sample of size 15 is taken from a larger population; the sample mean is calculated as 12 and the sample variance as 25. What is the 95% confidence interval for the population mean µ?

25 Example (page 8) We know that the confidence interval is given by x ± t p/2 s 2 /n, where

26 Example (page 8) We know that the confidence interval is given by x ± t p/2 s 2 /n, where n = 15,

27 Example (page 8) We know that the confidence interval is given by x ± t p/2 s 2 /n, where n = 15, x = 12 and

28 Example (page 8) We know that the confidence interval is given by x ± t p/2 s 2 /n, where n = 15, x = 12 s 2 = 25. and Also, to find t, we know that ν = n 1 = 15 1 = 14 and

29 Example (page 8) We know that the confidence interval is given by x ± t p/2 s 2 /n, where n = 15, x = 12 s 2 = 25. and Also, to find t, we know that ν = n 1 = 15 1 = 14 p = 5%. and

30 Example (page 8) We can find our t value by looking in the p = 5% column and the ν = 14 row, giving a value of Putting what we know into our expression, we get ± t 2.5% 15

31 Example (page 8) We can find our t value by looking in the p = 5% column and the ν = 14 row, giving a value of Putting what we know into our expression, we get ± t 2.5% ± i.e.

32 Example (page 8) We can find our t value by looking in the p = 5% column and the ν = 14 row, giving a value of Putting what we know into our expression, we get ± t 2.5% ± ± i.e.

33 Example (page 8) We can find our t value by looking in the p = 5% column and the ν = 14 row, giving a value of Putting what we know into our expression, we get ± t 2.5% ± ± i.e. Hence, the confidence interval is (9.23, 14.77).

34 Write this down! (Bottom page 8) It is claimed that µ = 9. Is this justified?

35 Write this down! (Bottom page 8) It is claimed that µ = 9. Is this justified? No! The claimed value of 9 does NOT lie within (9.23,14.77).

36 Confidence intervals: a general approach We now summarise the general procedure for calculating a confidence interval for the population mean µ.

37 Confidence intervals: a general approach We now summarise the general procedure for calculating a confidence interval for the population mean µ. Case 1: Known population variance σ 2 (i) Calculate the sample mean x from the data;

38 Confidence intervals: a general approach We now summarise the general procedure for calculating a confidence interval for the population mean µ. Case 1: Known population variance σ 2 (i) Calculate the sample mean x from the data; (ii) Calculate your interval! For example,

39 Confidence intervals: a general approach We now summarise the general procedure for calculating a confidence interval for the population mean µ. Case 1: Known population variance σ 2 (i) Calculate the sample mean x from the data; (ii) Calculate your interval! For example, for a 90% confidence interval, use the formula x ±1.645 σ 2 /n;

40 Confidence intervals: a general approach We now summarise the general procedure for calculating a confidence interval for the population mean µ. Case 1: Known population variance σ 2 (i) Calculate the sample mean x from the data; (ii) Calculate your interval! For example, for a 90% confidence interval, use the formula x ±1.645 σ 2 /n; for a 95% confidence interval, use the formula x ±1.96 σ 2 /n;

41 Confidence intervals: a general approach We now summarise the general procedure for calculating a confidence interval for the population mean µ. Case 1: Known population variance σ 2 (i) Calculate the sample mean x from the data; (ii) Calculate your interval! For example, for a 90% confidence interval, use the formula x ±1.645 σ 2 /n; for a 95% confidence interval, use the formula x ±1.96 σ 2 /n; for a 99% confidence interval, use the formula x ±2.576 σ 2 /n.

42 Confidence intervals: a general approach Case 2: Unknown population variance σ 2

43 Confidence intervals: a general approach Case 2: Unknown population variance σ 2 (i) Calculate the sample mean x and the sample variance s 2 from the data;

44 Confidence intervals: a general approach Case 2: Unknown population variance σ 2 (i) Calculate the sample mean x and the sample variance s 2 from the data; (ii) For a 100(1 p)% confidence interval, look up the value of t under column p, row ν of table 1.1, remembering that ν = n 1. Note that, for a 90% confidence interval, p = 10%, for a 95% confidence interval, p = 5% and for a 99% confidence interval, p = 1%;

45 Confidence intervals: a general approach Case 2: Unknown population variance σ 2 (i) Calculate the sample mean x and the sample variance s 2 from the data; (ii) For a 100(1 p)% confidence interval, look up the value of t under column p, row ν of table 1.1, remembering that ν = n 1. Note that, for a 90% confidence interval, p = 10%, for a 95% confidence interval, p = 5% and for a 99% confidence interval, p = 1%; (iii) Calculate your interval, using x ±t p/2 s 2 /n.

46 Application of Confidence Intervals You might be asking: why do we bother calculating confidence intervals?.

47 Application of Confidence Intervals You might be asking: why do we bother calculating confidence intervals?. By calculating a confidence interval for the population mean, it allows us to see how confident we are of the point estimate we have calculated. The wider the range, the less precise we can be about the population value.

48 Application of Confidence Intervals You might be asking: why do we bother calculating confidence intervals?. By calculating a confidence interval for the population mean, it allows us to see how confident we are of the point estimate we have calculated. The wider the range, the less precise we can be about the population value. If we have a known (or target) value for a population and this does not fall within the confidence interval of our sample, this could suggest that there is something different about this sample.

49 Application of Confidence Intervals You might be asking: why do we bother calculating confidence intervals?. By calculating a confidence interval for the population mean, it allows us to see how confident we are of the point estimate we have calculated. The wider the range, the less precise we can be about the population value. If we have a known (or target) value for a population and this does not fall within the confidence interval of our sample, this could suggest that there is something different about this sample. It allows us to start looking at differences between groups. If the confidence intervals for two samples do not overlap, this could suggest that they are from separate populations.

50 Example (page 9) A credit card company wants to determine the mean income of its card holders. It also wants to find out if there are any differences in mean income between males and females.

51 Example (page 9) A random sample of 225 male card holders and 190 female card holders was drawn, and the following results obtained: Mean Standard deviation Males Females Calculate 95% confidence intervals for the mean income for males and females. Is there any evidence to suggest that, on average, males and females incomes differ? If so, describe this difference.

52 Example (page 9) 95% confidence interval for male income The true population variance, σ 2, is unknown, and so we have case 2 and need to use the t distribution. Thus,

53 Example (page 9) 95% confidence interval for male income The true population variance, σ 2, is unknown, and so we have case 2 and need to use the t distribution. Thus, x ±t p/2 s 2 /n.

54 Example (page 9) 95% confidence interval for male income The true population variance, σ 2, is unknown, and so we have case 2 and need to use the t distribution. Thus, x ±t p/2 s 2 /n. Here, x = 16450, s 2 = = n = 225. and

55 Example (page 9) The value t p/2 must be found from table 1.1.

56 Example (page 9) The value t p/2 must be found from table 1.1. Recall that the degrees of freedom, ν = n 1, and so here we have ν = = 224;

57 Example (page 9) The value t p/2 must be found from table 1.1. Recall that the degrees of freedom, ν = n 1, and so here we have ν = = 224; But table 1.1 only gives value of ν up to 29; for higher values, we use the row;

58 Example (page 9) The value t p/2 must be found from table 1.1. Recall that the degrees of freedom, ν = n 1, and so here we have ν = = 224; But table 1.1 only gives value of ν up to 29; for higher values, we use the row; Since we require a 95% confidence interval, we read down the 5% column, giving a t value of 1.96.

59 Example (page 9) Thus, the 95% confidence interval for µ is found as

60 Example (page 9) Thus, the 95% confidence interval for µ is found as ± /225, i.e.

61 Example (page 9) Thus, the 95% confidence interval for µ is found as ± /225, i.e ±

62 Example (page 9) Thus, the 95% confidence interval for µ is found as ± /225, i.e ± So, the 95% confidence interval is ( , ).

63 Example (page 10) 95% confidence interval for female income Again, the true population variance, σ 2, is unknown, and so we have case 2. Thus,

64 Example (page 10) 95% confidence interval for female income Again, the true population variance, σ 2, is unknown, and so we have case 2. Thus, x ±t p/2 s 2 /n.

65 Example (page 10) 95% confidence interval for female income Again, the true population variance, σ 2, is unknown, and so we have case 2. Thus, x ±t p/2 s 2 /n. Now, x = 13220, s 2 = = , and n = 190.

66 Example (page 10) Again, since the sample size is large, we use the row of table 1.1 to obtain the value of t p/2, giving:

67 Example (page 10) Again, since the sample size is large, we use the row of table 1.1 to obtain the value of t p/2, giving: ± /190, i.e.

68 Example (page 10) Again, since the sample size is large, we use the row of table 1.1 to obtain the value of t p/2, giving: ± /190, i.e ± , i.e.

69 Example (page 10) Again, since the sample size is large, we use the row of table 1.1 to obtain the value of t p/2, giving: ± /190, i.e ± , i.e ±

70 Example (page 10) Again, since the sample size is large, we use the row of table 1.1 to obtain the value of t p/2, giving: ± /190, i.e ± , i.e ± So, the 95% confidence interval is ( , ).

71 Example (page 10) Since the 95% confidence intervals for males and females do not overlap, there is evidence to suggest that males and females incomes, on average, are different. Further, it appears that male card holders earn more than women. But note that the dataset is rather old...