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4. (a) Suppose we roll a dice 100 times and would like to determine if this dice is fair (null hypothesis). Say we compute the empirical mean and compare it to the mean of a fair dice. If we were to use the Z-test with significance level a = 0.05, how large or small does have to be for us to reject the null hypothesis? Express your answer as mathematical inequalities, and then use R to compute the answer numerically. Additional guidance: To use the Z-test we need to transform this into something that can be approximate by a normal distribution. If X₁,..., Xn are n = 100 fair dice rolls, then by the central limit theorem, Z:= 1 √n Xk - 3.5 σ 1 Here o is the standard deviation of one dice roll (calculate it!). The empirical Z is then xk-3.5 n ≈N(0, 1). k=1 Now, you should be able to use the Z-test... (b) Suppose we have a loaded dice where the probabilities are shown in the following table: x 1 2 3 4 5 6 P(X = x)| 1/4 1/8 1/8 1/8 1/8 1/4 Generate N = 105 experiments where each one consists of rolling this loaded dice 100 times. Of the N experiments, how often does the generated data fail the Z-test with significance value a = 0.05? (c) Same thing as the previous part with a different loaded dice: x 1 2 3 4 5 6 P(X = x)| 1/8 1/8 1/4 1/4 1/8 1/8 Report your results. (d) Compare your results for the two different loaded dice. Do they make intuitive sense and why?

4. (a) Suppose we roll a dice 100 times and would like to determine if this dice is fair (null hypothesis). Say we compute the empirical mean and compare it to the mean of a fair dice. If we were to use the Z-test with significance level a = 0.05, how large or small does have to be for us to reject the null hypothesis? Express your answer as mathematical inequalities, and then use R to compute the answer numerically. Additional guidance: To use the Z-test we need to transform this into something that can be approximate by a normal distribution. If X₁,..., Xn are n = 100 fair dice rolls, then by the central limit theorem, Z:= 1 √n Xk - 3.5 σ 1 Here o is the standard deviation of one dice roll (calculate it!). The empirical Z is then xk-3.5 n ≈N(0, 1). k=1 Now, you should be able to use the Z-test... (b) Suppose we have a loaded dice where the probabilities are shown in the following table: x 1 2 3 4 5 6 P(X = x)| 1/4 1/8 1/8 1/8 1/8 1/4 Generate N = 105 experiments where each one consists of rolling this loaded dice 100 times. Of the N experiments, how often does the generated data fail the Z-test with significance value a = 0.05? (c) Same thing as the previous part with a different loaded dice: x 1 2 3 4 5 6 P(X = x)| 1/8 1/8 1/4 1/4 1/8 1/8 Report your results. (d) Compare your results for the two different loaded dice. Do they make intuitive sense and why?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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4. (a) Suppose we roll a dice 100 times and would like to determine if this dice is fair (null hypothesis). Say we compute the empirical mean and compare it to the mean of a fair dice. If we were to use the Z-test with significance level a = 0.05, how large or small does have to be for us to reject the null hypothesis? Express your answer as mathematical inequalities, and then use R to compute the answer numerically. Additional guidance: To use the Z-test we need to transform this into something that can be approximate by a normal distribution. If X₁,..., Xn are n = 100 fair dice rolls, then by the central limit theorem, Z:= n 1 X P(X = x) √n k=1 Xk - 3.5 σ Here is the standard deviation of one dice roll (calculate it!). The empirical Z is then n 1 k=1 ≈N (0, 1). xk - 3.5 σ Now, you should be able to use the Z-test... (b) Suppose we have a loaded dice where the probabilities are shown in the following table: 1 2 3 4 5 6 1/4 1/8 1/8 1/8 1/8 1/4 Generate N = 105 experiments where each one consists of rolling this loaded dice 100 times. Of the N experiments, how often does the generated data fail the Z-test with significance value a = 0.05? (c) Same thing as the previous part with a different loaded dice: X 1 2 3 4 5 6 P(X= x) 1/8 1/8 1/4 1/4 1/8 1/8 Report your results. (d) Compare your results for the two different loaded dice. Do they make intuitive sense and why?
Transcribed Image Text:4. (a) Suppose we roll a dice 100 times and would like to determine if this dice is fair (null hypothesis). Say we compute the empirical mean and compare it to the mean of a fair dice. If we were to use the Z-test with significance level a = 0.05, how large or small does have to be for us to reject the null hypothesis? Express your answer as mathematical inequalities, and then use R to compute the answer numerically. Additional guidance: To use the Z-test we need to transform this into something that can be approximate by a normal distribution. If X₁,..., Xn are n = 100 fair dice rolls, then by the central limit theorem, Z:= n 1 X P(X = x) √n k=1 Xk - 3.5 σ Here is the standard deviation of one dice roll (calculate it!). The empirical Z is then n 1 k=1 ≈N (0, 1). xk - 3.5 σ Now, you should be able to use the Z-test... (b) Suppose we have a loaded dice where the probabilities are shown in the following table: 1 2 3 4 5 6 1/4 1/8 1/8 1/8 1/8 1/4 Generate N = 105 experiments where each one consists of rolling this loaded dice 100 times. Of the N experiments, how often does the generated data fail the Z-test with significance value a = 0.05? (c) Same thing as the previous part with a different loaded dice: X 1 2 3 4 5 6 P(X= x) 1/8 1/8 1/4 1/4 1/8 1/8 Report your results. (d) Compare your results for the two different loaded dice. Do they make intuitive sense and why?
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