Question

1. model with math coach henderson clocked the speeds in miles per hour of pitches thrown during the first inning of a middle school baseball game, as shown at the right. draw a box plot to display the data and write two conclusions about the data shown in the box plot. speeds of pitches thrown (in miles per hour) 45.3 47 48.1 51.3 55.8 61.1 48.5 60.7 49

Explanation:

Step1: Organize the data

First, we list the data points: \( 45.3, 47, 48.1, 48.5, 49, 51.3, 55.8, 60.7, 61.1 \) (Wait, let's check the count. Wait, the original data as per the image: let's re - list. Wait, maybe I missed some. Wait, the data given: 45.3, 47, 48.1, 51.3, 55.8, 61.1, 48.5, 60.7, 49. Let's sort them in ascending order:

Sorting the data: \( 45.3, 47, 48.1, 48.5, 49, 51.3, 55.8, 60.7, 61.1 \)

Number of data points \( n = 9 \) (odd).

Step2: Find the median (Q2)

The median is the middle value. For \( n = 9 \), the median is the \( \frac{n + 1}{2}=\frac{9+1}{2}=5^{th} \) term.

The sorted data: 1st: 45.3, 2nd: 47, 3rd: 48.1, 4th: 48.5, 5th: 49, 6th: 51.3, 7th: 55.8, 8th: 60.7, 9th: 61.1. So median \( Q2=49 \).

Step3: Find Q1 (lower quartile)

Q1 is the median of the lower half of the data. The lower half is the data points below the median. The lower half has \( \frac{n - 1}{2}=4 \) data points? Wait, for \( n = 9 \), the lower half is the first 4 data points? No, when \( n \) is odd, we exclude the median for finding Q1 and Q3. So lower half: \( 45.3, 47, 48.1, 48.5 \) (4 data points). The median of these 4 (even number) is the average of the 2nd and 3rd terms.

2nd term: 47, 3rd term: 48.1. So \( Q1=\frac{47 + 48.1}{2}=\frac{95.1}{2}=47.55 \)

Step4: Find Q3 (upper quartile)

The upper half of the data (excluding the median) is \( 51.3, 55.8, 60.7, 61.1 \). The median of these 4 data points is the average of the 2nd and 3rd terms.

2nd term: 55.8, 3rd term: 60.7. So \( Q3=\frac{55.8+60.7}{2}=\frac{116.5}{2}=58.25 \)

Step5: Find the minimum and maximum

Minimum value \( = 45.3 \), Maximum value \( = 61.1 \)

Step6: Draw the box - plot

• Draw a number line that covers the range from 45 to 62.
• Plot the minimum (45.3), Q1 (47.55), median (49), Q3 (58.25), and maximum (61.1) on the number line.
• Draw a box from Q1 to Q3. Inside the box, draw a vertical line at the median. Then draw whiskers from the box to the minimum and maximum values.

Step7: Write conclusions

Conclusion 1: The median speed of the pitches is 49 miles per hour, which means that half of the pitches have a speed less than or equal to 49 mph and half have a speed greater than or equal to 49 mph.

Conclusion 2: The inter - quartile range (IQR = Q3 - Q1) is \( 58.25−47.55 = 10.7 \) miles per hour. This IQR represents the spread of the middle 50% of the pitch speeds, indicating that the middle half of the pitch speeds vary by about 10.7 mph. Also, the minimum speed is 45.3 mph and the maximum is 61.1 mph, so the overall range of pitch speeds is \( 61.1 - 45.3=15.8 \) mph.

Answer:

The box - plot is drawn with minimum = 45.3, Q1 = 47.55, median = 49, Q3 = 58.25, maximum = 61.1. Conclusions: 1. Half of the pitches have speed ≤ 49 mph and half ≥ 49 mph. 2. The middle 50% of pitch speeds have a spread (IQR) of 10.7 mph.