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

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Presentation on theme: "Statistics Review – Part I Topics – Z-values – Confidence Intervals – Hypothesis Testing – Paired Tests – T-tests – F-tests 1."— Presentation transcript:

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


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