Review. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda

Transcription

1 Review DS GA 1002 Statistical and Mathematical Models Carlos Fernandez-Granda

2 Probability and statistics Probability: Framework for dealing with uncertainty Statistics: Framework for extracting information from data making probabilistic assumptions

3 Probability Probability basics: probability spaces, conditional probability, independence, conditional independence Random variables: pmf, cdf, pdf, important distributions, functions of random variables Multivariate random variables: joint pmf, joint cdf, joint pdf, marginal distributions, conditional distributions, independence, joint distribution of discrete/continuous random variables

4 Probability Expectation: definition, mean, median, variance, Markov and Chebyshev inequalities, covariance, correlation coefficient, covariance matrix, conditional expectation Random processes: definition, mean, autocovariance, important processes (iid, Gaussian, Poisson, random walk), Markov chains Convergence: types of convergence, law of large numbers, central limit theorem, convergence of Markov chains Simulation: motivation, inverse-transform sampling, rejection sampling, Markov-chain Monte Carlo

5 Statistics Descriptive statistics: histogram, empirical mean/variance, order statistics, empirical covariance, principal component analysis Statistical estimation: frequentist perspective, mean square error, consistency, confidence intervals Learning models: method of moments, maximum likelihood, empirical cdf, kernel density estimation

6 Statistics Hypothesis testing: definitions (null/alternative hypothesis, Type I/II errors), significance level, power, p value, parametric testing, power function, likelihood-ratio test, permutation test, multiple testing, Bonferroni s method Bayesian statistics: prior, likelihood, posterior, posterior mean/mode Linear regression: linear models, least squares, geometric interpretation, probabilistic interpretation, overfitting

7 Random walk with a drift We define the random walk X as the discrete-state discrete-time random process X (0) := 0, X (i) := X (i 1) + S (i) + 1, i = 1, 2,... where S (i) = { +1 with probability 1 2, 1 with probability 1 2, is an iid sequence of steps

8 Random walk with a drift What is the mean of this random process? ( ) E X (i)

9 Random walk with a drift What is the mean of this random process? ( ) i ( ) E X (i) = E S (j) + 1 j=1

10 Random walk with a drift What is the mean of this random process? ( ) i ( ) E X (i) = E S (j) + 1 = i j=1 j=1 ( ) E S (j) + n

11 Random walk with a drift What is the mean of this random process? ( ) i ( ) E X (i) = E S (j) + 1 = = i i j=1 j=1 ( ) E S (j) + n

12 Random walk with a drift What is the autocovariance? Use the fact that the autocovariance of the random walk without drift W that we studied in the lecture notes is R W (i, j) = min {i, j}

13 Random walk with a drift X (i)

14 Random walk with a drift X (i) = W (i) + i

15 Random walk with a drift ) E ( W (i) X (i) = W (i) + i

16 Random walk with a drift ) E ( W (i) X (i) = W (i) + i = 0

17 Random walk with a drift ) E ( W (i) X (i) = W (i) + i = 0 R X (i, j)

18 Random walk with a drift ) E ( W (i) X (i) = W (i) + i = 0 ( ) ( ) ( ) R X (i, j) := E X (i) X (j) E X (i) E X (j)

19 Random walk with a drift ) E ( W (i) X (i) = W (i) + i = 0 ( ) R X (i, j) := E X (i) X (j) = E ( ) E X (i) ( ) E X (j) (( W (i) + i ) ( W (j) + j )) E ) ) ( W (i) + i E ( W (j) + j

20 Random walk with a drift ) E ( W (i) X (i) = W (i) + i = 0 ( ) R X (i, j) := E X (i) X (j) ( ) E X (i) ) )) (( W (i) + i ( W (j) + j ( ) E X (j) ) = E E ( W (i) + i ) ) ) = E ( W (i) W (j) + ie ( W (j) + je ( W (i) + ij ie ( W (j) ) je ( W (i) ) ij ) E ( W (j) + j

21 Random walk with a drift ) E ( W (i) X (i) = W (i) + i = 0 ( ) R X (i, j) := E X (i) X (j) ( ) E X (i) ) )) (( W (i) + i ( W (j) + j ( ) E X (j) ) = E E ( W (i) + i ) ) ) = E ( W (i) W (j) + ie ( W (j) + je ( W (i) + ij ie = min {i, j} ( W (j) ) je ( W (i) ) ij ) E ( W (j) + j

22 Random walk with a drift Compute the first-order pmf of X (i). Recall that the first-order pmf of the random walk W equals {( i ) 1 i+x if i + x is even and i x i p W (i) (x) = 2 2 i 0 otherwise

23 Random walk with a drift p X (i) (x)

24 Random walk with a drift ( ) p X (i) (x) = P X (i) = x

25 Random walk with a drift ( ) p X (i) (x) = P X (i) = x ) = P ( W (i) = x i

26 Random walk with a drift ( ) p X (i) (x) = P X (i) = x ) = P ( W (i) = x i = p W (i) (x 1)

27 Random walk with a drift ( ) p X (i) (x) = P X (i) = x ) = P ( W (i) = x i = p W (i) (x 1) { =

28 Random walk with a drift ( ) p X (i) (x) = P X (i) = x ) = P ( W (i) = x i = p W (i) (x 1) = {( ix ) i

29 Random walk with a drift ( ) p X (i) (x) = P X (i) = x ) = P ( W (i) = x i = p W (i) (x 1) {( ix ) 1 if x is even and 0 x 2i 2 i = 2

30 Random walk with a drift ( ) p X (i) (x) = P X (i) = x ) = P ( W (i) = x i = p W (i) (x 1) {( ix ) 1 if x is even and 0 x 2i 2 i = 2 0 otherwise

31 Random walk with a drift Does the process satisfy the Markov condition? p X (i+1) X (1), X (2),..., X (i) (x i+1 x 1, x 2,..., x i ) = p X (i+1) X (i) (x i+1 x i )

32 Random walk with a drift p X (i+1) X (1), X (2),..., X (i) (x i+1 x 1, x 2,..., x i )

33 Random walk with a drift p X (i+1) X (1), X (2),..., X (i) (x i+1 x 1, x 2,..., x i ) = P (x i + S ) (i + 1) + 1 = x i+1

34 Random walk with a drift p X (i+1) X (1), X (2),..., X (i) (x i+1 x 1, x 2,..., x i ) = P (x i + S ) (i + 1) + 1 = x i+1 = p X (i+1) X (i) (x i+1 x i )

35 Random walk with a drift We observe that X (10) = 16 and X (20) = 30. What is the best estimator for X (21) in terms of probability of error?

36 Random walk with a drift p X (21) X (10), X (20) (x 16, 30)

37 Random walk with a drift p X (21) X (10), X (20) (x 16, 30) = p X (21) X (20) (x 30)

38 Random walk with a drift p X (21) X (10), X (20) (x 16, 30) = p X (21) X (20) (x 30) 1 2 if x = 32 = 1 2 if x = 30 0 otherwise

39 Markov chain Consider a Markov chain X with transition matrix [ ] a 1 T X :=, 1 a 0 where a is a constant between 0 and 1. We label the two states 0 and 1. The transition matrix T X has two eigenvectors q 1 := [ 1 ] [ ] 1 a 1, q 1 2 := 1 The corresponding eigenvalues are λ 1 := 1 and λ 2 := a 1

40 Markov chain For what values of a is the Markov chain irreducible?

41 Markov chain For what values of a is the Markov chain periodic?

42 Markov chain Express the stationary distribution of X in terms of a p stat

43 Markov chain Express the stationary distribution of X in terms of a p stat = 1 ( q 1 ) 1 + ( q 1 ) 2 q 1

44 Markov chain Express the stationary distribution of X in terms of a 1 p stat = q 1 ( q 1 ) 1 + ( q 1 ) 2 = 1 [ ] 1 2 a 1 a

45 Markov chain Does the Markov chain always converge in probability for all values of a? Justify that this is the case or provide a counterexample.

46 Markov chain Express the conditional pmf of X (i) conditioned on X (1) = 0 as a function of a and i. (Hint: Computing q 1 + q 2 could be a helpful first step.) Evaluate the expression at a = 0 and a = 1. Does the result make sense?

47 Markov chain We have q 1 + q 2

48 Markov chain We have q 1 + q 2 = [ 1 ] [ ] 1 a

49 Markov chain We have q 1 + q 2 = = [ 1 1 a ] [ ] [ 2 a 1 a 0 ]

50 Markov chain We have q 1 + q 2 = = [ 1 1 a 1 [ 2 a 1 a 0 ] [ ] ] p X (0)

51 Markov chain We have q 1 + q 2 = = [ 1 1 a 1 [ 2 a 1 a 0 ] [ ] ] p X (0) = [ ] 1 0

52 Markov chain We have q 1 + q 2 = = [ 1 1 a 1 [ 2 a 1 a 0 ] [ ] ] p X (0) = [ ] 1 0 = 1 a 2 a ( q 1 + q 2 )

53 Markov chain p X (i)

54 Markov chain p X (i) = T ĩ X p X (0)

55 Markov chain p X (i) = T ĩ p X X (0) = T ĩ 1 a X 2 a ( q 1 + q 2 )

56 Markov chain p X (i) = T ĩ p X X (0) = T ĩ 1 a X 2 a ( q 1 + q 2 ) = 1 a ( λ i 2 a 1 q 1 + λ i 2 q 2)

57 Markov chain p X (i) = T ĩ p X X (0) = T ĩ 1 a X 2 a ( q 1 + q 2 ) = 1 a ( λ i 2 a 1 q 1 + λ i 2 q 2) = 1 a 2 a ([ 1 ] [ ]) 1 a + (a 1) i 1 1 1

58 Markov chain p X (i) = T ĩ p X X (0) = T ĩ 1 a X 2 a ( q 1 + q 2 ) = 1 a ( λ i 2 a 1 q 1 + λ i 2 q 2) = 1 a 2 a = 1 2 a ([ 1 1 a ] [ ]) + (a 1) i 1 1 ] 1 [ 1 (a 1) i+1 (1 a) (1 (a 1) i)

59 Markov chain For a = 1 we have p X (i) = [ ] 1 0

60 Markov chain For a = 0 we have p X (i) = 1 2 [ ] 1 ( 1) i+1 1 ( 1) i

61 Markov chain For a = 0 we have [ ] p X (i) = 1 1 ( 1) i ( 1) i [ ] 0 if i is odd, 1 = [ ] 1 if i is even. 0

62 Sampling from multivariate distributions We are interested in generating samples from the joint distribution of two random variables X and Y. If we generate a sample x according to the pdf f X and a sample y according to the pdf f Y, are these samples a realization of the joint distribution of X and Y? Explain your answer with a simple example.

63 Sampling from multivariate distributions Now, assume that X is discrete and Y is continuous. Propose a method to generate a sample from the joint distribution using the pmf of X and the conditional cdf of Y given X using two independent samples from a distribution that is uniform between 0 and 1. Assume that the conditional cdf is invertible.

64 Sampling from multivariate distributions 1. Obtain two independent samples u 1 and u 2 from the uniform distribution.

65 Sampling from multivariate distributions 1. Obtain two independent samples u 1 and u 2 from the uniform distribution. 2. Set x to equal the smallest value a such that p X (a) 0 and u 1 F X (a).

66 Sampling from multivariate distributions 1. Obtain two independent samples u 1 and u 2 from the uniform distribution. 2. Set x to equal the smallest value a such that p X (a) 0 and u 1 F X (a). 3. Define Set y := F 1 x (u 2 ) F x ( ) := F Y X ( x)

67 Sampling from multivariate distributions Explain how to generate samples from a random variable with pdf f W (w) = 0.1 λ 1 exp ( λ 1 w) λ 2 exp ( λ 2 w), w 0, where λ 1 and λ 2 are positive constants, using two iid uniform samples between 0 and 1.

68 Sampling from multivariate distributions Let us define a Bernoulli random variable X with parameter 0.9, such that if X = 0 then Y is exponential with parameter λ 1 and if X = 1 then Y is exponential with parameter λ 2

69 Sampling from multivariate distributions Let us define a Bernoulli random variable X with parameter 0.9, such that if X = 0 then Y is exponential with parameter λ 1 and if X = 1 then Y is exponential with parameter λ 2 The marginal distribution of Y is f Y (w) = p X (0) f Y X (w 0) + p X (1) f Y X (w 1)

70 Sampling from multivariate distributions Let us define a Bernoulli random variable X with parameter 0.9, such that if X = 0 then Y is exponential with parameter λ 1 and if X = 1 then Y is exponential with parameter λ 2 The marginal distribution of Y is f Y (w) = p X (0) f Y X (w 0) + p X (1) f Y X (w 1) = 0.1 λ 1 exp ( λ 1 w) λ 2 exp ( λ 2 w)

71 Sampling from multivariate distributions 1. We obtain two independent samples u 1 and u 2 from the uniform distribution.

72 Sampling from multivariate distributions 1. We obtain two independent samples u 1 and u 2 from the uniform distribution. 2. If u we set w := 1 ( ) 1 log λ 1 1 u 2 otherwise we set w := 1 ( ) 1 log λ 2 1 u 2

73 Convergence Let U be a random variable uniformly distributed between 0 and 1. If we define the discrete random process X X (i) = U for all i, does X converge to 1 U in probability?

74 Convergence Does X converge to 1 U in distribution?

75 Convergence You draw some iid samples x 1, x 2,... from a Cauchy random variable. Will the empirical mean 1 n n i=1 x i converge in probability as n grows large? Explain why briefly and if the answer is yes state what it converges to.

76 Convergence You draw m iid samples x 1, x 2,..., x m from a Cauchy random variable. Then you draw iid samples y 1, y 2,... uniformly from {x 1, x 2,..., x m } (each y i is equal to each element of {x 1, x 2,..., x m } with probability 1/m). Will the empirical mean 1 n n i=1 y i converge in probability as n grows large? Explain why very briefly and if the answer is yes state what it converges to.

77 Earthquake We are interested in learning a model for the occurrence of earthquakes. We decide to model the time between earthquakes as an exponential random variable with parameter λ. Compute the maximum-likelihood estimate of λ given t 1, t 2,..., t n, which are interarrival times for past earthquakes. Assume that the data are iid.

78 Earthquake L (λ)

79 Earthquake L (λ) := f T (1),..., T (n) (t 1,..., t n )

80 Earthquake L (λ) := f T (1),..., T (n) (t 1,..., t n ) n = λ exp ( λt i ) i=1

81 Earthquake L (λ) := f T (1),..., T (n) (t 1,..., t n ) n = λ exp ( λt i ) i=1 ( = λ n exp λ ) n t i i=1

82 Earthquake L (λ) := f T (1),..., T (n) (t 1,..., t n ) n = λ exp ( λt i ) i=1 ( = λ n exp λ ) n t i i=1 log L (λ)

83 Earthquake L (λ) := f T (1),..., T (n) (t 1,..., t n ) n = λ exp ( λt i ) i=1 ( = λ n exp λ ) n t i i=1 log L (λ) = n log λ λ n i=1 t i

84 Earthquake d log L t1,...,t n (λ) dλ

85 Earthquake d log L t1,...,t n (λ) dλ = n λ n i=1 t i

86 Earthquake d log L t1,...,t n (λ) dλ = n λ n i=1 t i d 2 log L t1,...,t n (λ) dλ 2

87 Earthquake d log L t1,...,t n (λ) dλ = n λ n i=1 t i d 2 log L t1,...,t n (λ) dλ 2 = n λ 2

88 Earthquake d log L t1,...,t n (λ) dλ = n λ n i=1 t i d 2 log L t1,...,t n (λ) dλ 2 = n λ 2 λ ML

89 Earthquake d log L t1,...,t n (λ) dλ = n λ n i=1 t i d 2 log L t1,...,t n (λ) dλ 2 = n λ 2 1 λ ML = 1 n n i=1 t i

90 Earthquake Find an approximate 0.95 confidence interval based on the central limit theorem for the value of λ. Assume that you know a bound b on the standard deviation (i.e. the variance of the exponential 1/λ 2 is bounded by b 2 ) and express your answer using the Q function. (Hint: Express the ML estimate in terms of the empirical mean.) (See solutions.)

91 Earthquake What is the posterior distribution of the parameter Λ if we model it as a random variable with a uniform distribution between 0 and u? Express your answer in terms of the sum n i=1 t i, u and the marginal pdf of the data evaluated at t 1, t 2,..., t n c := f T (1),..., T (n) (t 1,..., t n ).

92 Earthquake f Λ T (1),..., T (n) (λ t 1,..., t n )

93 Earthquake f Λ T (1),..., T (n) (λ t 1,..., t n ) = f Λ (λ) λ n exp ( λ n i=1 t i) f T (1),..., T (n) (t 1,..., t n )

94 Earthquake f Λ T (1),..., T (n) (λ t 1,..., t n ) = f Λ (λ) λ n exp ( λ n i=1 t i) f T (1),..., T (n) (t 1,..., t n ) = 1 u c λn exp ( λ ) n t i i=1

95 Earthquake f Λ T (1),..., T (n) (λ t 1,..., t n ) = f Λ (λ) λ n exp ( λ n i=1 t i) f T (1),..., T (n) (t 1,..., t n ) = 1 u c λn exp ( λ for 0 λ u and zero otherwise ) n t i i=1

96 Earthquake f Λ T (1),..., T (n) (λ t1,..., tn) λ

97 Earthquake Explain how you would use the answer in the previous question to construct a confidence interval for the parameter

98 Chad You hate a coworker and want to predict when he is in the office from the temperature. Chad No Chad You model his presence using a random variable C which is equal to 1 if he is there and 0 if he is not. Estimate p C.

99 Chad The empirical pmf is p C (0) = 5 15 = 1 3, p C (1) = = 2 3.

100 Chad You model the temperature using a random variable T. Sketch the kernel density estimator of the conditional distribution of T given C using a rectangular kernel with width equal to 2.

101 Chad 0.20 f T C (t 0) 0.15 f T C (t 1)

102 Chad If T = 68 what is the ML estimate of C?

103 Chad If T = 68 what is the ML estimate of C? f T C (68 0) = 0.2 f T C (68 1) = 0

104 Chad If T = 64 what is the MAP estimate of C?

105 Chad If T = 64 what is the MAP estimate of C? p C T (0 64)

106 Chad If T = 64 what is the MAP estimate of C? p C (0) f T C (64 0) p C T (0 64) = p C (0) f T C (64 0) + p C (1) f T C (64 1)

107 Chad If T = 64 what is the MAP estimate of C? p C (0) f T C (64 0) p C T (0 64) = p C (0) f T C (64 0) + p C (1) f T C (64 1) =

108 Chad If T = 64 what is the MAP estimate of C? p C (0) f T C (64 0) p C T (0 64) = p C (0) f T C (64 0) + p C (1) f T C (64 1) = =

109 Chad If T = 64 what is the MAP estimate of C? p C (0) f T C (64 0) p C T (0 64) = p C (0) f T C (64 0) + p C (1) f T C (64 1) = = p C T (1 64)

110 Chad If T = 64 what is the MAP estimate of C? p C (0) f T C (64 0) p C T (0 64) = p C (0) f T C (64 0) + p C (1) f T C (64 1) = = p C T (1 64) = 1 p C T (0 64)

111 Chad If T = 64 what is the MAP estimate of C? p C (0) f T C (64 0) p C T (0 64) = p C (0) f T C (64 0) + p C (1) f T C (64 1) = = p C T (1 64) = 1 p C T (0 64) = 1 2

112 Chad What happens if the temperature is 57? Explain how using parametric estimation may alleviate this problem.

113 3-point shooting The New York Knicks hire you as a data analyst. Your first task is to come up with a way to determine whether a 3-point shooter is any good. You will use the following graph of the function g (θ, n) = θ n. g(θ,n) n = 4 n = 9 n = 14 n = 19 n = θ

114 3-point shooting 1. Interpret g (θ, n).

115 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis?

116 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%?

117 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? 14

118 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter?

119 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? 0.14

120 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter?

121 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) 0.76

122 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) You apply the test on 10 players. You adapt the threshold applying Bonferroni s method. What is the new threshold?

123 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) You apply the test on 10 players. You adapt the threshold applying Bonferroni s method. What is the new threshold? n = 24

124 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) You apply the test on 10 players. You adapt the threshold applying Bonferroni s method. What is the new threshold? n = With the correction, what is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter?

125 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) You apply the test on 10 players. You adapt the threshold applying Bonferroni s method. What is the new threshold? n = With the correction, what is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) 0.92

126 3-point shooting 1. Interpret g (θ, n). 2. The coach tells you: I want to make sure that the guy has a shooting percentage over 80%. What is your null hypothesis? 3. What number of shots does a player need to make in a row for you to reject the null hypothesis with a confidence level of 5%? A player makes 9 shots in a row. What is the corresponding p value? Do you declare him as a good shooter? What is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) You apply the test on 10 players. You adapt the threshold applying Bonferroni s method. What is the new threshold? n = With the correction, what is the probability that you do not declare a player who has a shooting percentage of 90% as a good shooter? 1 g (0.9, 14) What is the advantage of adapting the threshold? What is the disadvantage?