Aim for FiveStatistics

Question 4: Multi-focus question

Pizza delivery times

  • Units 2, 3 and 4
  • 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 E, and label any subparts (for example, i and ii).

The delivery claim

A pizza chain advertises that its delivery times have a mean of 30 minutes and a standard deviation of 8 minutes. The distribution of delivery times is skewed to the right, because a few deliveries take much longer than usual.

Source: Hypothetical scenario

Suggested time: 22 minutes

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Part (A)

1 point

Explain why it would not be appropriate to use a normal distribution to calculate the probability that a single randomly selected delivery takes more than 35 minutes.

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Part (B)

2 points

Consider the mean delivery time, x̄, for a random sample of 40 deliveries. Assume the advertised mean and standard deviation are correct. (i) Describe the shape of the sampling distribution of x̄. Justify your answer. (ii) Calculate the mean and standard deviation of the sampling distribution of x̄.

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Part (C)

1 point

Assuming the advertised values are correct, calculate the probability that the mean delivery time for a random sample of 40 deliveries is greater than 32 minutes.

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Part (D)

5 points

A consumer group selected a random sample of 40 of the chain's deliveries from last month. The sample mean was 32.9 minutes and the sample standard deviation was 9.1 minutes. The group wants to test, at the α = 0.05 significance level, whether the chain's mean delivery time last month was greater than 30 minutes. Assume the 10% condition is met. (i) Identify the appropriate inference procedure. (ii) State the hypotheses, and define the parameter. (iii) Calculate the test statistic and p-value. (iv) Compare the p-value with α, and make a decision about the null hypothesis. (v) State your conclusion in context.

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Part (E)

1 point

Describe a Type I error in the context of the consumer group's test.

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