Aim for FiveStatistics

Question 2: Analyzing data and interpreting results

Juice bottle fill amounts

  • Units 1 and 2
  • 10 points
  • About 22 minutes

You can use a calculator on this question, just like on exam day.

A multi-part question built on a data set or summary statistics: you calculate things like outlier fences, compare distributions, pick the right measure of center or spread, and justify a claim in context. It focuses on Practices 3 and 4. On the exam: Question 2 of 4 (official name: Multi-Focus on Practices 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 H, and label any subparts (for example, i and ii).

The filling machine

A machine fills juice bottles labeled 500 milliliters (mL). The amount of juice it puts in a bottle is approximately normally distributed with mean 502 mL and standard deviation 4 mL. Assume the fill amounts of different bottles are independent.

Source: Hypothetical scenario

Suggested time: 22 minutes

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Part (A)

1 point

Calculate the proportion of bottles that contain less than 500 mL. Show your work.

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Part (B)

1 point

Calculate the fill amount at the 90th percentile of the distribution. Show your work.

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Part (C)

1 point

Calculate the proportion of bottles that contain between 495 mL and 510 mL. Show your work.

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Part (D)

1 point

Use the empirical rule to find the interval of fill amounts that contains about the middle 95% of bottles.

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Part (E)

1 point

Two bottles are selected at random. Calculate the probability that both contain less than 500 mL.

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Part (F)

2 points

A quality inspector finds a bottle that contains 490 mL. (i) Calculate the z-score for this bottle. (ii) Explain whether this bottle suggests that the machine may not be working as described.

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Part (G)

2 points

A case holds 24 bottles. Let X be the number of bottles in a randomly selected case that contain less than 500 mL. (i) Calculate the mean and standard deviation of X. (ii) Calculate the probability that at least 10 bottles in the case contain less than 500 mL.

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Part (H)

1 point

Interpret the mean of X in context.

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