Probability distributions

Paper labels are an unofficial, independently authored grouping, applied only where a question's own format genuinely matches a real paper convention (such as IB Mathematics AA's non-calculator/calculator split). They do not reproduce any exam board's real paper numbering or mark allocation, and uncertain questions are labelled general practice instead.

• 1Customers arrive at a small bakery counter at an average rate of 5 per 10-minute interval, and the number of arrivals follows a Poisson distribution. (a) State the mean number of arrivals in a 10-minute interval. (b) Calculate the probability that exactly 3 customers arrive in a randomly chosen 10-minute interval. (c) Calculate the probability that at least one customer arrives in a 2-minute interval.Paper 1 styleMedium4 marksCalculator+
• 2Marking analysis: A learner attempts the following task: “Customers arrive at a small bakery counter at an average rate of 5 per 10-minute interval, and the number of arrivals follows a Poisson distribution. (a) State the mean number of arrivals in a 10-minute interval. (b) Calculate the probability that exactly 3 customers arrive in a randomly chosen 10-minute interval. (c) Calculate the probability that at least one customer arrives in a 2-minute interval.” Their response addresses only this point: “States λ = 5 for a 10-minute interval.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium4 marksNo calculator+
• 3The heights of a large population of a plant species are normally distributed with mean 42 cm and standard deviation 6 cm. (a) Calculate the probability that a randomly selected plant is taller than 50 cm. (b) Calculate the probability that a randomly selected plant has a height between 36 cm and 48 cm. (c) The tallest 10% of plants are labelled 'premium'. Find the minimum height for a plant to be labelled 'premium'.Paper 2 styleMedium5 marksCalculator+
• 4Marking analysis: A learner attempts the following task: “The heights of a large population of a plant species are normally distributed with mean 42 cm and standard deviation 6 cm. (a) Calculate the probability that a randomly selected plant is taller than 50 cm. (b) Calculate the probability that a randomly selected plant has a height between 36 cm and 48 cm. (c) The tallest 10% of plants are labelled 'premium'. Find the minimum height for a plant to be labelled 'premium'.” Their response addresses only this point: “Uses the normal distribution with mean 42 and standard deviation 6.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisMedium5 marksNo calculator+
• 5A researcher believes a new fertiliser increases the mean mass of tomatoes. A random sample of 36 tomatoes grown with the fertiliser has a mean mass of 182 g with a sample standard deviation of 24 g. Tomatoes grown without the fertiliser have a known population mean mass of 172 g. Test, at the 5% significance level, whether the sample provides evidence that the fertiliser increases the mean mass. (a) State the null and alternative hypotheses. (b) Calculate the test statistic. (c) State the p-value for this test. (d) State a conclusion in context, referring to the significance level. (e) State one assumption needed for this test to be valid.Paper 3 style (extended)Hard7 marksCalculator+
• 6Marking analysis: A learner attempts the following task: “A researcher believes a new fertiliser increases the mean mass of tomatoes. A random sample of 36 tomatoes grown with the fertiliser has a mean mass of 182 g with a sample standard deviation of 24 g. Tomatoes grown without the fertiliser have a known population mean mass of 172 g. Test, at the 5% significance level, whether the sample provides evidence that the fertiliser increases the mean mass. (a) State the null and alternative hypotheses. (b) Calculate the test statistic. (c) State the p-value for this test. (d) State a conclusion in context, referring to the significance level. (e) State one assumption needed for this test to be valid.” Their response addresses only this point: “States H0: μ = 172 g.” Evaluate the response against the complete 7-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisHard7 marksNo calculator+