Question 4: Multi-focus question
Taco truck orders
- Units 2 and 3
- 10 points
- About 22 minutes
You can use a calculator on this question, just like on exam day.
A multi-part question that pulls together several units, such as probability from a table, a summary statistic and an inference procedure, and asks you to calculate, interpret and justify a claim. It uses Practices 2, 3 and 4. On the exam: Question 4 of 4 (official name: Multi-Focus on Practices 2, 3, and 4). Section II has 4 free-response questions in 90 minutes (50% of the score); each question is worth 10 points and 12.5% of the score. The exam is fully digital in Bluebook, a graphing calculator with statistical capabilities is expected, and the formula sheet and tables are provided.
The question and its sources
Show the work that leads to each answer, and explain your reasoning in the context of the situation. A graphing calculator with statistical capabilities, the formula sheet and the statistical tables are available. Use the information given to answer parts A through D, and label any subparts (for example, i and ii).
The taco truck
A food truck sells tacos, and each order includes from 1 to 4 tacos. Let X be the number of tacos in a randomly selected order. Based on months of sales records, the owner gives the probability distribution of X in the table.
Source: Hypothetical scenario
Probability distribution of X
| Number of tacos, x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X = x) | 0.15 | 0.35 | 0.30 | 0.20 |
Source: Hypothetical data
Suggested time: 22 minutes
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Part (A)
3 points(i) Calculate the probability that a randomly selected order includes at least 3 tacos. (ii) Calculate the mean of X. (iii) Calculate the standard deviation of X. Show your work for each calculation.
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Part (B)
1 pointInterpret the mean of X in context.
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Part (C)
2 pointsSuppose 25 orders are selected at random, and the number of tacos in each order is independent of the others. Let Y be the number of these 25 orders that include exactly 4 tacos. (i) Explain why Y has a binomial distribution, and give its parameters. (ii) Calculate the probability that at least 8 of the 25 orders include exactly 4 tacos.
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Part (D)
4 pointsThe owner adds a "4-taco combo" deal and wants to know whether the proportion of orders that include exactly 4 tacos has increased from 0.20. In a random sample of 150 orders after the deal began, 42 included exactly 4 tacos. Assume the conditions for inference are met. (i) Identify the appropriate inference procedure. (ii) State the hypotheses, and define the parameter. (iii) Calculate the test statistic and p-value. (iv) At the α = 0.05 significance level, justify a conclusion in context.
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