Math / Data & Statistics

QuestionThe accompanying data represent the wait time (in minutes) for a random sample of forty visitors to an amusement park ride. Complete parts (a) and (b). \begin{tabular}{llllllllll} 19 & 1 & 2 & 24 & 5 & 11 & 5 & 9 & 9 & 13 \\ 7 & 9 & 33 & 9 & 3 & 11 & 6 & 45 & 3 & 25 \\ 5 & 17 & 16 & 32 & 9 & 61 & 17 & 7 & 8 & 9 \\ 10 & 4 & 19 & 5 & 13 & 6 & 21 & 7 & 12 & 4 \end{tabular} (a) Determine and interpret the quartiles.
By the quartiles, about $\square$ $\%$ of the wait times are $\mathrm{Q}_{1}=$ $\square$ minute(s) or less, and about $\square$ $\%$ of the wait times exceed $Q_{1}$ minute(s); about $\square$ $\%$ of the wait times are $\mathrm{Q}_{2}=$ $\square$ minute(s) 0 $\square$ $\%$ of the wait times are less and about $\square$ $\%$ of the wait times exceed $\mathrm{Q}_{2}$ minute(s); about $\square$ $\mathrm{Q}_{3}=$ minute(s) or less, and about $\square$ $\%$ of the wait times exceed $\mathrm{Q}_{3}$ minute(s). (Type integers or decimals. Do not round.)

Studdy Solution

STEP 1

1. The data consists of wait times for a sample of 40 visitors to an amusement park ride.
2. The quartiles (Q1, Q2, Q3) divide the data into four equal parts.
3. Q1 represents the 25th percentile, Q2 represents the 50th percentile (median), and Q3 represents the 75th percentile.

STEP 2

1. Organize the data in ascending order.
2. Determine the position of Q1 (25th percentile).
3. Determine the position of Q2 (50th percentile or median).
4. Determine the position of Q3 (75th percentile).
5. Interpret the quartiles in the context of the problem.

STEP 3

Organize the data in ascending order.
$\{1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 10, 11, 11, 12, 13, 13, 16, 17, 17, 19, 19, 21, 24, 25, 32, 33, 45, 61\}$

STEP 4

Determine the position of Q1 (25th percentile).
For a data set of size $n=40$ , the position of Q1 is given by:
$P_{Q1} = \frac{1}{4}(n+1) = \frac{1}{4}(40+1) = \frac{41}{4} = 10.25$
Q1 is the 10.25th position, so we interpolate between the 10th (5) and 11th (6) values:
$Q1 = 5 + 0.25 \times (6 - 5) = 5 + 0.25 = 5.25$

STEP 5

Determine the position of Q2 (50th percentile or median).
For a data set of size $n=40$ , the position of Q2 is given by:
$P_{Q2} = \frac{2}{4}(n+1) = \frac{2}{4}(40+1) = \frac{41}{2} = 20.5$
Q2 is the 20.5th position, so we interpolate between the 20th (9) and 21st (9) values:
$Q2 = 9 + 0.5 \times (9 - 9) = 9$

STEP 6

Determine the position of Q3 (75th percentile).
For a data set of size $n=40$ , the position of Q3 is given by:
$P_{Q3} = \frac{3}{4}(n+1) = \frac{3}{4}(40+1) = \frac{123}{4} = 30.75$
Q3 is the 30.75th position, so we interpolate between the 30th (16) and 31st (17) values:
$Q3 = 16 + 0.75 \times (17 - 16) = 16 + 0.75 = 16.75$

STEP 7

Interpret the quartiles in the context of the problem.
By the quartiles: - About 25% of the wait times are $\mathrm{Q}_1 = 5.25$ minutes or less, and about 75% of the wait times exceed $5.25$ minutes. - About 50% of the wait times are $\mathrm{Q}_2 = 9$ minutes or less, and about 50% of the wait times exceed $9$ minutes. - About 75% of the wait times are $\mathrm{Q}_3 = 16.75$ minutes or less, and about 25% of the wait times exceed $16.75$ minutes.
Solution: By the quartiles, about $25\%$ of the wait times are $\mathrm{Q}_1 = 5.25$ minute(s) or less, and about $75\%$ of the wait times exceed $Q_1$ minute(s); about $50\%$ of the wait times are $\mathrm{Q}_2 = 9$ minute(s) or less and about $50\%$ of the wait times exceed $\mathrm{Q}_2$ minute(s); about $75\%$ of the wait times are $\mathrm{Q}_3 = 16.75$ minute(s) or less, and about $25\%$ of the wait times exceed $\mathrm{Q}_3$ minute(s).
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