Question 2: Analyzing data and interpreting results

Practice time and free throws

Unit 5
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 I, and label any subparts (for example, i and ii).

The free-throw data

A basketball coach recorded, for each of the 14 players on her team, the number of hours of free-throw practice per week (x) and the percent of free throws the player made during the season (y). The practice times ranged from 2 to 14 hours per week, with a mean of x̄ = 7.5 hours.

The least-squares regression line is ŷ = 48.2 + 2.35x, where x is practice hours per week and ŷ is the predicted percent of free throws made. The correlation is r = 0.82.

A residual plot (residuals on the vertical axis, practice hours on the horizontal axis) shows the 14 residuals scattered above and below 0 with no curved or fan-shaped pattern.

Source: Hypothetical scenario

Suggested time: 22 minutes

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Part (A)

1 point

Describe the relationship between practice time and free-throw percentage. Use the information given.

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Part (B)

1 point

Interpret the slope of the least-squares regression line in context.

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Part (C)

1 point

Interpret the y-intercept of the least-squares regression line in context.

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Part (D)

2 points

One player practiced 6 hours per week and made 66% of free throws. (i) Calculate the residual for this player. Show your work. (ii) Interpret the residual in context.

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Part (E)

1 point

Calculate and interpret the coefficient of determination, r², in context.

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Part (F)

1 point

Explain how the residual plot supports using a linear model for these data.

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Part (G)

1 point

Use the regression line to predict the free-throw percentage for a player who practices 25 hours per week. Explain why this prediction should not be trusted.

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Part (H)

1 point

The coach says the data prove that more practice causes a higher free-throw percentage. Explain why the coach's claim is not justified by these data.

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Part (I)

1 point

Calculate the mean free-throw percentage, ȳ, for the 14 players. Explain how you found it.

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