Lecture 11. Data Description Estimation

Transcription

1 Lecture 11 Data Description Estimation

2 Measures of Central Tendency (continued, see last lecture) Sample mean, population mean Sample mean for frequency distributions The median The mode The midrange

3 3-22 The Weighted Mean The weighted mean is used when the values in a data set are not all equally represented. The weighted mean of a variable X is found by multiplying each value by its corresponding weight and dividing the sum of the products by the sum of the weights.

4 3-22 The Weighted Mean The weighted mean X = w X + w X w X n w + w w 1 2 wx = w where w, w,..., w are the weights 1 2 for the values X, X,..., X. n 1 2 n n n

5 Distribution Shapes Frequency distributions can assume many shapes. The three most important shapes are positively skewed,, symmetrical, and negatively skewed.

6 Positively Skewed Y Positively Skewed Mode < Median < Mean X

7 Symmetrical Y Symmetrical Mean = Media n = Mode X

8 Negatively Skewed Y Negatively Skewed Mean < Median < Mode X

9 3-33 Measures of Variation - Range The range is defined to be the highest value minus the lowest value. The symbol R is used for the range. R = highest value lowest value. Extremely large or extremely small data values can drastically affect the range.

10 3-33 Measures of Variation - Population Variance The variance is the average of the squares of the distance each value is from the mean. The symbol for the population variance is 2 σ ( σ is the Greek lowercase letter sigma) 2 2 ( X µ ) σ =, where N X = individual value µ = population mean N = population size

11 3-33 Measures of Variation - Population Standard Deviation The standard deviation is the square root of the variance. σ = 2 σ = 2 ( X µ ). N

12 3-33 Measures of Variation - Example Consider the following data to constitute the population: 10, 60, 50, 30, 40, 20. Find the mean and variance. The mean µ = ( )/6 = 210/6 = 35. The variance σ 2 = 1750/6 = See next slide for computations.

13 3-33 Measures of Variation - Example X X -- µ (( X -- µ 2 ) 2 )

14 3-33 Measures of Variation - Sample Variance The unbiased estimator of the population variance or the sample variance is a statistic whose value approximates the expected value of a population variance. It is denoted by s 2, where 2 2 ( X X ) s =, and n 1 X = sample mean n = sample size

15 3-33 Measures of Variation - Sample Standard Deviation The sample standard deviation is the square root of the sample variance. s = s 2 = 2 ( X X ). n 1

16 3-3 Shortcut Formula for the Sample Variance and the Standard Deviation s 2 = 2 2 X ( X ) / n n 1 s = 2 2 X ( X ) / n n 1

17 3-33 Sample Variance - Example Find the variance and standard deviation for the following sample: 16, 19, 15, 15, 14. ΣX = = 79. ΣX 2 = = 1263.

18 3-33 Sample Variance - Example s X ( X ) / n = n (79) / 5 = = s = 3.7 = 19..

19 3-33 Sample Variance for Grouped and Ungrouped Data For grouped data, use the class midpoints for the observed value in the different classes. For ungrouped data, use the same formula (see next slide) with the class midpoints, X m, replaced with the actual observed X value.

20 3-33 Sample Variance for Grouped and Ungrouped Data The samplevariance for grouped data: s 2 = 2 2 f X m [( f X m) / n]. n 1 For ungrouped data, replace X m with the observed X value.

21 3-33 Sample Variance for Grouped Data - Example X ff f X f X f X f X n = Σf X = Σf X 2 2 = 1500

22 3-33 Sample Variance for Ungrouped Data - Example The sample variance and standard deviation: s 2 2 f X [( f X ) / n] 2 s = n [(186)2/ 24] = = = = 16..

23 3-33 Coefficient of Variation The coefficient of variation is defined to be the standard deviation divided by the mean. The result is expressed as a percentage. s CVar= 100% or CVar = σ 100%. X µ

24 The Empirical (Normal) Rule For any bell shaped distribution: Approximately 68% of the data values will fall within one standard deviation of the mean. Approximately 95% will fall within two standard deviations of the mean. Approximately 99.7% will fall within three standard deviations of the mean.

25 The Empirical (Normal) Rule µ ± 1σ 68% µ ± 2σ -- 95% µ ± 3σ 99.7% µ 3σ µ 2σ µ 1σ µ µ +1σ µ +2σ µ +3σ

26 3-44 Measures of Position z score The standard score or z score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation. The symbol z is used for the z score.

27 3-44 Measures of Position z-score The z score represents the number of standard deviations a data value falls above or below the mean. For samples: X X z = s For populations: X µ z =. σ.

28 3-44 z-score z - Example A student scored 65 on a statistics exam that had a mean of 50 and a standard deviation of 10. Compute the z-score. z = (65 50)/10 = 1.5. That is, the score of 65 is 1.5 standard deviations above the mean. Above - since the z-score is positive.

29 3-44 Quartiles Quartiles divide the data set into 4 groups. Quartiles are denoted by Q 1, Q 2, and Q 3. The median is the same as Q 2.

30 3-44 Outliers and the Interquartile Range (IQR) An outlier is an extremely high or an extremely low data value when compared with the rest of the data values. The Interquartile Range, IQR = Q 3 Q. 1

31 3-55 Exploratory Data Analysis - Box Plot When the data set contains a small number of values, a box plot is used to graphically represent the data set. These plots involve five values: the minimum value, the lower hinge (Q( 1 ), the median, the upper hinge (Q( 3 ), and the maximum value.

32 3-55 Exploratory Data Analysis - Box Plot The lower hinge is the other name for the lower quartile Q 1, denoted by the symbol LH. The upper hinge is the other name for the upper quartile Q 3, denoted by the symbol UH.

33 Exploratory Data Analysis - Box Plot - Example (Cardiograms data) LH UH MINIMUM MAXIMUM MEDIAN

34 Information Obtained from a Box Plot If the median is near the center of the box, the distribution is approximately symmetric. If the median falls to the left of the center of the box, the distribution is positively skewed. If the median falls to the right of the center of the box, the distribution is negatively skewed.

35 Information Obtained from a Box Plot If the lines are about the same length, the distribution is approximately symmetric. If the right line is larger than the left line, the distribution is positively skewed. If the left line is larger than the right line, the distribution is negatively skewed.

36 Estimations

37 Outline 8-1 Introduction 8-2 Confidence Intervals for the Mean [σ Known or n 30] and Sample Size 8-3 Confidence Intervals for the Mean [σ Unknown and n < 30]

38 8-1 Introduction One aspect of inferential statistics is estimation, which is the process of estimating the value of a parameter from information obtained from a sample. An important question: How large should the sample be in order to make an accurate estimate?

39 8-1 Introduction Two types of estimations: A point estimate is a specific numerical value estimate of a parameter. An interval estimate of a parameter is an interval or a range of values used to estimate the parameter.

40 8-2 Point estimates A point estimate is a specific numerical value estimate of a parameter. The best estimate of the population mean µ is the sample mean X.

41 8-22 Three Properties of a Good Estimator The estimator must be an unbiased estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated.

42 8-22 Three Properties of a Good Estimator The estimator must be consistent. For a consistent estimator, as sample size increases, the value of the estimator approaches the value of the parameter estimated.

43 8-22 Three Properties of a Good Estimator The estimator must be a relatively efficient estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.

44 8-22 Confidence Intervals An interval estimate interval estimate of a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated.

45 8-22 Confidence Intervals A confidence interval confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and the specific confidence level of the estimate.

46 8-22 Confidence Intervals The confidence level confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter.

47 8-2 Confidence Intervals for the Mean (σ Known or n 30) and Sample Size Three common confidence intervals are used: the 90%, the 95%, and the 99% confidence intervals. When the sample size is large, approximately 95% of the sample means will fall within ±1.96 standard errors of the population mean:

48 8-2 Confidence Intervals for the Mean (σ Known or n 30) and Sample Size If a specific sample mean is selected, say, there is a 95% probability that it falls within the range of Likewise, there is a 95% probability that the interval specified by will contain µ.

49 8-2 Confidence Intervals for the Mean (σ Known or n 30) and Sample Size

50 8-22 Formula for the Confidence Interval of the Mean for a Specific α The confidence level is the percentage equivalent to the decimal value of 1 α. σ σ X z < µ < X + n α 2 α 2 z n

51 8-22 Maximum Error of Estimate The maximum error of estimate is the maximum difference between the point estimate of a parameter and the actual value of the parameter.

52 8-22 Confidence Intervals - Example The president of a large university wishes to estimate the average age of the students presently enrolled. From past studies, the standard deviation is known to be 2 years. A sample of 50 students is selected, and the mean is found to be 23.2 years. Find the 95% confidence interval of the population mean.

53 8-22 Confidence Intervals - Example Since the 95% confidence interval is desired, z = Hence, α 2 substituting in the formula σ X zα < µ < X + z n one gets 2 α 2 σ n

54 8-22 Confidence Intervals - Example (1.96)( ) < µ < (1.96)( ) < µ < < µ < 238. or 23.2 ± 0.6 years. Hence, the president can say, with 95% confidence, that the average age of the students is between and 238. years, based on 50 students.

55 8-22 Confidence Intervals - Example A certain medication is known to increase the pulse rate of its users. The standard deviation of the pulse rate is known to be 5 beats per minute. A sample of 30 users had an average pulse rate of 104 beats per minute. Find the 99% confidence interval of the true mean.

56 8-22 Confidence Intervals - Example Since the 99% confidence interval is desired, z = Hence, α 2 substituting in the formula σ X z < µ < α X + z n one gets 2 α 2 σ n

57 8-22 Confidence Intervals - Example (2.58). ( ) < µ < (2.58)( ) < µ < < µ < Hence, one can say, with 99% confidence, that the average pulse rate is between and beats per minute, based on 30 users.

58 The meaning of 95% confidence

59 8-2 Formula for the Minimum Sample Size Needed for an Interval Estimate of the Population Mean n = of estimate. z α where E is the σ 2 E If necessary, round the answer up to obtain a whole number. 2 maximum error

60 8-22 Minimum Sample Size Needed for an Interval Estimate of the Population Mean - Example The college president asks the statistics teacher to estimate the average age of the students at their college. How large a sample is necessary? The statistics teacher decides the estimate should be accurate within 1 year and be 99% confident. From a previous study, the standard deviation of the ages is known to be 3 years.

61 8-22 Minimum Sample Size Needed for an Interval Estimate of the Population Mean - Example Since α = ( or ), z = 2. 58, and E = 1, substituting in n n α 2 = z α σ 2 E 2 gives = ( )( 3 ) 2 =

62 8-33 Characteristics of the t Distribution The t distribution shares some characteristics of the normal distribution and differs from it in others. The t distribution is similar to the standard normal distribution in the following ways: It is bell-shaped. It is symmetrical about the mean.

63 8-33 Characteristics of the t Distribution The mean, median, and mode are equal to 0 and are located at the center of the distribution. The curve never touches the x axis. The t distribution differs from the standard normal distribution in the following ways:

64 8-33 Characteristics of the t Distribution The variance is greater than 1. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to the sample size. As the sample size increases, the t distribution approaches the standard normal distribution.

65 8-33 Standard Normal Curve and the t Distribution

66 8-33 Confidence Interval for the Mean (σ Unknown and n < 30) - Example Ten randomly selected automobiles were stopped, and the tread depth of the right front tire was measured. The mean was 0.32 inch, and the standard deviation was 0.08 inch. Find the 95% confidence interval of the mean depth. Assume that the variable is approximately normally distributed.

67 8-33 Confidence Interval for the Mean (σ Unknown and n < 30) - Example Since σ is unknown and s must replace it, the t distribution must be used with α = Hence, with 9 degrees of freedom, t α/2 = (see Table F in text). From the next slide, we can be 95% confident that the population mean is between 0.26 and 0.38.

68 8-33 Confidence Interval for the Mean (σ Unknown and n < 30) - Example Thus the confidence of the population mean is found by substituting in X t 95% s < µ < X + t n α 2 α 2 interval s n 0.32 (2.262) 0.08 < µ < (2.262) < µ < 0. 38

69 When to use the z or t distribution?