Decision Probability Worksheets Grade 11

1. A pharmaceutical company is testing a new drug and claims it reduces recovery time by at least 2 days compared to the standard treatment. The null hypothesis states H₀: μ = 2 (mean reduction is 2 days), while the alternative hypothesis states H₁: μ < 2 (mean reduction is less than 2 days). After conducting the trial with a sample of patients, researchers calculate a p-value of 0.032. Using a significance level of α = 0.05, should the company reject the null hypothesis and conclude the drug is less effective than claimed?
2. Aroha is evaluating two investment options. Option A: gain $3200 with probability 0.30, lose $800 with probability 0.70. Option B: gain $2100 with probability 0.50, lose $600 with probability 0.50. Which option has the higher expected value and by how much?
3. Liam is evaluating two investment strategies. Strategy A: gain $3500 with probability 0.25, lose $900 with probability 0.75. Strategy B: gain $2100 with probability 0.45, lose $500 with probability 0.55. Which strategy has the higher expected value and by how much?

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1. A medical researcher is testing whether a new physical therapy protocol improves mobility scores for patients recovering from knee surgery. The null hypothesis states there is no improvement (H₀: μ = 0), while the alternative hypothesis claims there is improvement (H₁: μ > 0). After conducting a one-sample t-test with 36 patients, the researcher obtains a test statistic of t = 2.42. Using a significance level of α = 0.01 and knowing the critical value for a one-tailed test with 35 degrees of freedom is approximately 2.438, what decision should the researcher make about the therapy's effectiveness?
2. A city is evaluating whether to implement a new traffic light system at a dangerous intersection. The current system has an average of 4.2 accidents per month. After installing the new system for 6 months, the city recorded the following monthly accident counts: 3, 2, 4, 1, 3, 2. The city wants to test if the new system significantly reduces accidents using a one-tailed t-test with α = 0.05. The null hypothesis is H₀: μ = 4.2 (no improvement), and the alternative is H₁: μ < 4.2. What is the calculated t-statistic, and should the city reject the null hypothesis?
3. A medical researcher is testing whether a new therapy reduces anxiety levels in patients. She conducts a hypothesis test with H₀: μ = 50 (no change in anxiety score) and H₁: μ < 50 (anxiety decreases). With a sample of 36 patients, she calculates a test statistic of t = -2.45. Using a significance level of α = 0.01 and knowing the critical value for a one-tailed test with 35 degrees of freedom is approximately -2.438, should she reject the null hypothesis?

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1. Charlotte is a video game developer deciding between two monetization strategies for her new mobile game. In Strategy A, players pay a flat fee of $12 to download the game, with a 0.25 probability that a player will make an additional in-app purchase averaging $20, a 0.45 probability of a smaller purchase averaging $8, and a 0.30 probability of no further purchase. In Strategy B, the game is free to download, but players are shown ads; each player generates $3.50 in ad revenue, but there is a 0.15 probability that the ads will drive the player away before they generate any revenue, a 0.60 probability they stay and generate the $3.50, and a 0.25 probability they become a premium subscriber paying $15. Based on expected revenue per player, which strategy should Charlotte choose, and what is the expected revenue for that strategy?
2. Matiu is a data analyst for a logistics company. He is evaluating two different shipping strategies for delivering packages to a remote region. Strategy A uses a single large truck that has a 15% chance of being delayed by weather, resulting in a $1,200 penalty for late delivery, and an 85% chance of arriving on time with no penalty. The base cost of operating this truck is $4,500. Strategy B uses two smaller vans. Van 1 has a 10% chance of a mechanical issue costing $800 in repairs and a 90% chance of running smoothly with no extra cost. Van 2 has a 25% chance of taking a wrong route costing $600 in extra fuel and a 75% chance of taking the correct route with no extra cost. The base cost of operating both vans together is $3,200. Matiu must choose the strategy with the lower total expected cost (base cost plus expected penalties/repairs/fuel). Which strategy should Matiu choose, and what is its total expected cost?
3. Emma is deciding between two investment options for a $5000 bonus she received. Option A is a low-risk bond that guarantees a profit of $250 with probability 0.90, but has a 0.10 probability of losing $50 due to early withdrawal penalties. Option B is a high-risk stock that has a 0.35 probability of gaining $600, a 0.50 probability of gaining $100, and a 0.15 probability of losing $300. Which investment option has the higher expected monetary value, and what is that expected value?

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