Download presentation

Presentation is loading. Please wait.

Published byAllyson Broom Modified over 2 years ago

1
Statistics Review – Part I Topics – Z-values – Confidence Intervals – Hypothesis Testing – Paired Tests – T-tests – F-tests 1

2
Statistics References References used in class slides: 1.Sullivan III, Michael. Statistics: Informed Decisions Using Data, Pearson Education, Gitlow, et. al Six Sigma for Green Belts and Champions, Prentice Hall,

3
3 Relative frequency histograms that are symmetric and bell- shaped are said to have the shape of a normal curve. Sampling and the Normal Distribution

4
4 If a continuous random variable is normally distributed or has a normal probability distribution, then a relative frequency histogram of the random variable has the shape of a normal curve (bell-shaped and symmetric). Sampling and the Normal Distribution

5
5

6
Suppose that the mean normal sugar level in the population is 0 =9.7mmol/L with std. dev. =2.0mmol/L - you want to see whether diabetics have increased blood sugar level Sample n=64 individuals with diabetes mean is 0 =13.7mmol/L with std. dev. =2.0mmol/L How do you compare these values? – Standardize! 6 Sampling and the Normal Distribution

7
7 Reading z-scores Sampling and the Normal Distribution

8
Standardization: – Using Z-tables to evaluate sample means – Puts samples on the same scale Subtract mean and divide by standard deviation 8 Sampling and the Normal Distribution

9
Why do we standardize? – Enables the comparison of populations/ samples using a standardized set of values – Recall 9 Sampling and the Normal Distribution

10
10 The table gives the area under the standard normal curve for values to the left of a specified Z-score, z o, as shown in the figure. Sampling and the Normal Distribution

11
11 Sampling and the Normal Distribution

12
– Population Mean=10, Standard Deviation=5 – What is the likelihood of a sample (n=16) having a mean greater than 12 (standard deviation = 5)? – What is the likelihood of a sample (n=16) having a mean of less than 8 (standard deviation = 5)? 12 Sampling and the Normal Distribution

13
13 Notation for the Probability of a Standard Normal Random Variable: P(a < Z < b) represents the probability a standard normal random variable is between a and b P(Z > a)represents the probability a standard normal random variable is greater than a. P(Z < a) represents the probability a standard normal random variable is less than a. Sampling and the Normal Distribution

14
Before using Z-tables, need to assess whether the data is normally distributed Different ways – Histogram – Probability plot 14 Sampling and the Normal Distribution

15
15 Normal Probability Plots: Sampling and the Normal Distribution

16
16 Normal Probability Plots: Fat pencil test to detect normality Sampling and the Normal Distribution

17
17 Sampling and the Normal Distribution Shapes of Normal Probability Plots:

18
18 Sampling and the Normal Distribution Normal Probability Plots vs Box plots:

19
If distribution of data is “approximately” normally distributed, use Z-tables to determine likelihood of events 19 Sampling and the Normal Distribution

20
Can also “flip” Z-scores to determine the ‘highest’ or ‘lowest’ acceptable sample mean 20 Sampling and the Normal Distribution

21
– Point estimate: value of a statistic that estimates the value of the parameter. – Confidence interval estimate: interval of numbers along with a probability that the interval contains the unknown parameter. – Level of confidence: a probability that represents the percentage of intervals that will contain if a large number of repeated samples are obtained. 21 Confidence Intervals

22
22 The construction of a confidence interval for the population mean depends upon three factors: The point estimate of the population The level of confidence The standard deviation of the sample mean: A 95% level if 100 confidence intervals were constructed, each based on a different sample from the same population, we would expect 95 of the intervals to contain the population mean. Confidence Intervals

23
23 If a simple random sample from a population is normally distributed or the sample size is large, the distribution of the sample mean will be normal with: Confidence Intervals

24
24 Confidence Intervals

25
25 95% of all sample means are in the interval: With a little algebraic manipulation, we can rewrite this inequality and obtain: Confidence Intervals

26
26 Confidence Intervals

27
27 Steps to constructing a confidence interval: 1.Verify normality if n<=30. 2.Determine /2, x-bar, . 3.Find z-score for /2. 4.Calculate upper and lower bound. Confidence Intervals

28
28 Confidence Intervals

29
29 Confidence Intervals

30
30 Histogram for z Confidence Intervals

31
31 Histogram for t Confidence Intervals

32
32 Confidence Intervals

33
33 Properties of the t Distribution 1.The t distribution is different for different values of n. 2. The t distribution is centered at 0 and is symmetric about The area under the curve is 1. The area under the curve to the right of 0 = the area under the curve to the left of 0 = 1 / As t increases and decreases without bound, the graph approaches, but never equals, zero. 5.The area in the tails of the t distribution is a little greater than the area in the tails of the standard normal distribution. This is due to using s as an estimate introducing more variability to the t statistic. 6.As the sample size n increases, the density of the curve of t approaches the standard normal density curve. The occurs due to the values of s approaching the values of sigma by the law of large numbers. Confidence Intervals

34
34 Confidence Intervals

35
35 Confidence Intervals

36
36 EXAMPLE: Finding t-values Find the t-value such that the area under the t distribution to the right of the t-value is 0.2 assuming 10 degrees of freedom. Hint: find t 0.20 with 10 degrees of freedom. Confidence Intervals

37
37 Confidence Intervals

38
38 Confidence Intervals

39
39 Confidence Intervals

40
40 Confidence Intervals

41
41 EXAMPLE: Finding Chi-Square Values Find the chi-square values that separate the middle 95% of the distribution from the 2.5% in each tail. Assume 18 degrees of freedom. Confidence Intervals

42
42 Confidence Intervals

43
43 EXAMPLE: Constructing a Confidence Interval about a Population Standard Deviation Confidence Intervals

44
44 Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of one or more populations. Selecting Hypothesis Testing methods – see next slides. Hypothesis Testing

45

46

47
47 The null hypothesis, denoted H o (read “H-naught”), is a statement to be tested. The null hypothesis is assumed true until evidence indicates otherwise. In this chapter, it will be a statement regarding the value of a population parameter. The alternative hypothesis, denoted, H 1 (read “H-one”), is a claim to be tested. We are trying to find evidence for the alternative hypothesis. In this chapter, it will be a claim regarding the value of a population parameter. Hypothesis Testing

48
48 There are three ways to set up the null and alternative hypothesis: 1. Equal versus not equal hypothesis (two-tailed test) H o : parameter = some value H 1 : parameter some value 2. Equal versus less than (left-tailed test) H o : parameter = some value H 1 : parameter < some value 3. Equal versus greater than (right-tailed test) H o : parameter = some value H 1 : parameter > some value Hypothesis Testing

49
49 THREE WAYS TO STRUCTURE THE HYPOTHESIS TEST: Hypothesis Testing

50
Two-tailed test 50 Hypothesis Testing

51
One-Tailed Test 51 Hypothesis Testing

52
52 Four Outcomes from Hypothesis Testing 1. We could reject H o when in fact H 1 is true. This would be a correct decision. 2. We could not reject H o when in fact H o is true. This would be a correct decision. 3. We could reject H o when in fact H o is true. This would be an incorrect decision. This type of error is called a Type I error. 4. We could not reject H o when in fact H 1 is true. This would be an incorrect decision. This type of error is called a Type II error. Hypothesis Testing

53
53 For example, we might reject the null hypothesis if the sample mean is more than 2 standard deviations above the population mean. Why? z Area =

54
54 If the null hypothesis is true, then = = 97.72% of all sample means will be less than Hypothesis Testing

55
55 Because sample means greater than 2.88 are unusual if the population mean is 2.62, we are inclined to believe the population mean is greater than Hypothesis Testing

56
56 Hypothesis Testing

57
57 Step 1: A claim is made regarding the population mean. The claim is used to determine the null and alternative hypotheses. Again, the hypothesis can be structured in one of three ways: Hypothesis Testing

58
58 Hypothesis Testing

59
59 The critical region or rejection region is the set of all values such that the null hypothesis is rejected. Hypothesis Testing

60
60 Hypothesis Testing

61
61 Step 4: Compare the critical value with the test statistic: Step 5: State the conclusion. Hypothesis Testing

62
62 A P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption the null hypothesis is true. Hypothesis Testing

63
63 Hypothesis Test Regarding μ with σ Known (P-values) Hypothesis Testing

64
64 Step 1: A claim is made regarding the population mean. The claim is used to determine the null and alternative hypotheses. Again, the hypothesis can be structured in one of three ways: Hypothesis Testing

65
65 Step 3: Compute the P-value. Hypothesis Testing

66
66 Hypothesis Testing

67
67 Hypothesis Testing

68
68 Hypothesis Testing

69
69 Properties of the t Distribution 1.The t distribution is different for different values of n, the sample size. 2.The t distribution is centered at 0 and is symmetric about 0. 3.The area under the curve is 1. Because of the symmetry, the area under the curve to the right of 0 equals the area under the curve to the left of 0 equals ½. 4.As t increases without bound, the graph approaches, but never equals, zero. As t decreases without bound the graph approaches, but never equals, zero. 5.The area in the tails of the t distribution is a little greater than the area in the tails of the standard normal distribution. This result is because we are using s as an estimate of which introduces more variability to the t statistic. Hypothesis Testing

70
70 Hypothesis Testing

71
71 Step 1: A claim is made regarding the population mean. The claim is used to determine the null and alternative hypotheses. Again, the hypothesis can be structured in one of three ways: Hypothesis Testing

72
72 Hypothesis Testing

73
73 Step 3: Compute the test statistic which follows Student’s t-distribution with n – 1 degrees of freedom. Hypothesis Testing

74
74 Step 4: Compare the critical value with the test statistic: Step 5 : State the conclusion. Hypothesis Testing

75
75 Hypothesis Testing

76
76 Hypothesis Testing

77
77 Hypothesis Test Regarding a Population Variance or Standard Deviation If a claim is made regarding the population variance or standard deviation, we can use the following steps to test the claim provided (1) the sample is obtained using simple random sampling (2) the population is normally distributed Hypothesis Testing

78
78 Step 1: A claim is made regarding the population variance or standard deviation. The claim is used to determine the null and alternative hypothesis. We present the three cases for a claim regarding a population standard deviation.

79
79 Hypothesis Testing

80
80 Hypothesis Testing

81
81 Step 4: Compare the critical value with the test statistic. Step 5: State the conclusion. Hypothesis Testing

82
82 A sampling method is independent when the individuals selected for one sample does not dictate which individuals are to be in a second sample. A sampling method is dependent when the individuals selected to be in one sample are used to determine the individuals to be in the second sample. Dependent samples are often referred to as matched pairs samples. Paired Testing

83
83 EXAMPLEIndependent versus Dependent Sampling For each of the following, determine whether the sampling method is independent or dependent. (a) A researcher wants to know whether the price of a one night stay at a Holiday Inn Express Hotel is less than the price of a one night stay at a Red Roof Inn Hotel. She randomly selects 8 towns where the location of the hotels is close to each other and determines the price of a one night stay. (b) A researcher wants to know whether the newly issued “state” quarters have a mean weight that is different from “traditional” quarters. He randomly selects 18 “state” quarters and 16 “traditional” quarters. Their weights are compared. Paired Testing

84
84 In order to test the hypotheses regarding the mean difference, we need certain requirements to be satisfied. A simple random sample is obtained The sample data is matched pairs The differences are normally distributed or the sample size, n, is large (n > 30). Paired Testing

85
85 Paired Testing

86
86 Paired Testing

87
87 Paired Testing

88
88 Step 4: Compare the critical value with the test statistic: Step 5 : State the conclusion. Paired Testing

89
89 T-Tests

90
90

91
91 T-Tests

92
92 T-Tests

93
93 T-Tests

94
94 T-Tests

95
95 Step 4: Compare the critical value with the test statistic: Step 5 : State the conclusion. T-Tests

96
96 The degrees of freedom used to determine the critical value(s) presented in the last example are conservative. Results that are more accurate can be obtained by using the following degrees of freedom: T-Tests

97
97 Lower Bound = Upper Bound =

98
98 Requirements for Testing Claims Regarding Two Population Standard Deviations 1. The samples are independent simple random samples. 2. The populations from which the samples are drawn are normally distributed. F-Tests

99
99

100
100 Fisher's F-distribution

101
101 Characteristics of the F-distribution 1. It is not symmetric. The F-distribution is skewed right. 2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the distribution and Student’s t-distribution, whose shape depends upon their degrees of freedom. 3. The total area under the curve is The values of F are always greater than or equal to zero. F-Tests

102
102 F-Tests

103
103 Is the critical F with n 1 – 1 degrees of freedom in the numerator and n 2 – 1 degrees of freedom in the denominator and an area of to the right of the critical F. To find the critical F with an area of α to the left, use the following: F-Tests

104
104 Hypothesis Test Regarding the Two Means Population Standard Deviations F-Tests

105
105 F-Tests

106
106 F-Tests

107
107 F-Tests

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google