3. (8 marks) In an effort to get more funding, two interns are giving chocolates to a pod of inverted lobsters to see what happen (if anything happens). Intern A found that eating chocolates had no effect on inverted lobsters' swimming ability. However, intern B found that eating chocolate before swimming improved performance by an average of 0.4 seconds per 100 metres covered in open water. Your task: Follow the following steps to calculate the probability that intern A's hypothesis is wrong assuming that intern B is correct. We are going to calculate P(false negatives). (a) (1 mark) In two or three sentences, explain why we should calculate the probability of false negatives of simulated data. (b) (1 mark) State and justify the criterion to determine if a simulated experiment (replica) should count as a false negative. (c) (1 mark) State and justify the uniform distribution U(a, ß) from which simulated data should be drawn. Approximate your numbers to two decimal places. Hint: You can use the following shortcut without justification: Find U(-a, a) using the required variance. It follows that a = µ - a and ẞ = μ+ a where μ is the required expectation. (d) (4 marks) Adapt the following code for Example 5 on page 121 of the course textbook to calculate the probability of intern A committing a type-II error using 200,000 simulated replicas of the original experiment (takes about 5 minutes on Colab). > H:::::::://colab.research.google.com/drive/1tUUwv8UZUbU4pA60Vs8f6zcJEu1VP794 • The code above contains some additional randomness in the pseudorandom number generator. Each run will give you a different result. This behaviour is expected. • Change num_replicas to something small, such as num_replicas = 10 to speed up run time while you develop your simulation. ⚫ The final run before your submission should have num_replicas = 200000. • See the submission guideline below. Intern A's original data is provided below. Copy and paste into your Colab notebook. # data from intern A's original experiment # (swimming performance enhancement measured in seconds per 100 metres in open water) original_data = [ -2.7936, -3.5744, 1.0497, -2.9049, -1.0128, 0.9132, 1.2411, -0.4170, 0.6782, 0.5905, -1.2312, -0.3636, -2.4040, -1.2662, 0.2434, 1.5055, -1.6589, 0.9094, -1.2683, -0.8885, -0.9140, -0.5514, -2.7295, 0.2749, -1.0202, -1.5952, 0.3119, -1.3676, -1.7045, -1.2771, -2.7500, -0.4590, -0.9361, -1.3172, 0.8075, -1.4469, -0.5031, 4.0849, -2.2451, -0.7821, -1.3970, 2.1970, 2.1557, -0.8235, 1.6479, 1.1594, -1.4729, 0.1700, -0.9560, -3.5777, -2.2417, 0.0223, 0.0078, 1.1708, 0.0796, 0.3316, -0.6491, 1.0230, 0.0564, -1.7434, 0.9106, 0.7952, 0.5020, -0.9738, 1.2751, -1.3273, -0.7157, -0.7731, 0.7443, 1.5110, -0.9877, -1.9794, 1.3925, -0.1988, 0.6328, 0.8226, -0.1642, -1.2415, -0.6105, 0.8763, -0.5499, 2.0696, -1.3863, 1.3777, -0.7738, -1.0966, 0.0053, -2.1853, -1.0489, 0.3654, -1.3999, -1.1090, -0.7452, 0.3560, -1.9975, 0.2243, -0.0535, -0.0185, 0.0786, 0.5432 ] Submission guideline. Submit the Colab notebook as a PDF. Make sure that you can Restart session and run all (see the screenshot below) without any errors before creating the PDF.

3. (8 marks) In an effort to get more funding, two interns are giving chocolates to a pod of inverted lobsters to see what happen (if anything happens). Intern A found that eating chocolates had no effect on inverted lobsters' swimming ability. However, intern B found that eating chocolate before swimming improved performance by an average of 0.4 seconds per 100 metres covered in open water. Your task: Follow the following steps to calculate the probability that intern A's hypothesis is wrong assuming that intern B is correct. We are going to calculate P(false negatives). (a) (1 mark) In two or three sentences, explain why we should calculate the probability of false negatives of simulated data. (b) (1 mark) State and justify the criterion to determine if a simulated experiment (replica) should count as a false negative. (c) (1 mark) State and justify the uniform distribution U(a, ß) from which simulated data should be drawn. Approximate your numbers to two decimal places. Hint: You can use the following shortcut without justification: Find U(-a, a) using the required variance. It follows that a = µ - a and ẞ = μ+ a where μ is the required expectation. (d) (4 marks) Adapt the following code for Example 5 on page 121 of the course textbook to calculate the probability of intern A committing a type-II error using 200,000 simulated replicas of the original experiment (takes about 5 minutes on Colab). > H:::::::://colab.research.google.com/drive/1tUUwv8UZUbU4pA60Vs8f6zcJEu1VP794 • The code above contains some additional randomness in the pseudorandom number generator. Each run will give you a different result. This behaviour is expected. • Change num_replicas to something small, such as num_replicas = 10 to speed up run time while you develop your simulation. ⚫ The final run before your submission should have num_replicas = 200000. • See the submission guideline below. Intern A's original data is provided below. Copy and paste into your Colab notebook. # data from intern A's original experiment # (swimming performance enhancement measured in seconds per 100 metres in open water) original_data = [ -2.7936, -3.5744, 1.0497, -2.9049, -1.0128, 0.9132, 1.2411, -0.4170, 0.6782, 0.5905, -1.2312, -0.3636, -2.4040, -1.2662, 0.2434, 1.5055, -1.6589, 0.9094, -1.2683, -0.8885, -0.9140, -0.5514, -2.7295, 0.2749, -1.0202, -1.5952, 0.3119, -1.3676, -1.7045, -1.2771, -2.7500, -0.4590, -0.9361, -1.3172, 0.8075, -1.4469, -0.5031, 4.0849, -2.2451, -0.7821, -1.3970, 2.1970, 2.1557, -0.8235, 1.6479, 1.1594, -1.4729, 0.1700, -0.9560, -3.5777, -2.2417, 0.0223, 0.0078, 1.1708, 0.0796, 0.3316, -0.6491, 1.0230, 0.0564, -1.7434, 0.9106, 0.7952, 0.5020, -0.9738, 1.2751, -1.3273, -0.7157, -0.7731, 0.7443, 1.5110, -0.9877, -1.9794, 1.3925, -0.1988, 0.6328, 0.8226, -0.1642, -1.2415, -0.6105, 0.8763, -0.5499, 2.0696, -1.3863, 1.3777, -0.7738, -1.0966, 0.0053, -2.1853, -1.0489, 0.3654, -1.3999, -1.1090, -0.7452, 0.3560, -1.9975, 0.2243, -0.0535, -0.0185, 0.0786, 0.5432 ] Submission guideline. Submit the Colab notebook as a PDF. Make sure that you can Restart session and run all (see the screenshot below) without any errors before creating the PDF.

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.7: Probability

Problem 26E: Find the probability of each event. Getting 2 red eggs in a single scoop from a bucket containing 5...

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