Question 4: Multi-focus question
Ice cream sales and temperature
- Units 4 and 5
- 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 ice cream shop
The owner of an ice cream shop selected a random sample of 20 summer weekdays and an independent random sample of 10 summer weekend days from the past ten summers. For each of the 30 days, she recorded the high temperature, in degrees Fahrenheit (°F), and the number of cones sold.
Using all 30 days, the least-squares regression line is ŷ = −112 + 4.6x, where x is the day's high temperature and ŷ is the predicted number of cones sold. The correlation is r = 0.91. The high temperatures ranged from 68°F to 97°F, and a residual plot showed no pattern.
The table summarizes the number of cones sold by type of day. Dotplots of the cones sold for each type of day show no strong skewness and no outliers.
Source: Hypothetical scenario
Cones sold per day
| Type of day | n | Mean | SD |
|---|---|---|---|
| Weekend | 10 | 297 | 41 |
| Weekday | 20 | 248 | 34 |
Source: Hypothetical data
Suggested time: 22 minutes
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Part (A)
1 pointInterpret the slope of the least-squares regression line in context.
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Part (B)
2 pointsOn one day in the sample, the high temperature was 85°F and the shop sold 290 cones. (i) Calculate the residual for this day. (ii) Interpret the residual in context.
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Part (C)
1 pointCalculate and interpret the coefficient of determination, r², in context.
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Part (D)
4 pointsThe owner wants to estimate how much more, on average, the shop sells on summer weekend days than on summer weekdays. (i) Identify the appropriate inference procedure. (ii) Assume the conditions for inference are met. Construct a 90% confidence interval. (iii) Interpret the interval in context.
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Part (E)
2 pointsThe owner claims that, on average, the shop sells more than 20 additional cones on a summer weekend day than on a summer weekday. (i) Determine whether the interval from part D supports the owner's claim. (ii) Explain your reasoning.
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