Write down the modal class of the data. Calculate the estimated mean weight-loss of the race car drivers…

Question

Write down the modal class of the data. Calculate the estimated mean weight-loss of the race car drivers. Sketch the ogive (cumulative frequency graph) representing race 2 in the ANSWER BOOK. How will the range of race 2 compare with the range of race 1? Determine the average weight lost in race 2.

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Here's the solution to Question 2: 2.1.1 Write down the modal class of the data. Step 1: Identify the class interval with the highest frequency. From the table, the highest frequency is 15, which corresponds to the interval 2500 ≤ w < 3000. The modal class is 2500 ≤ w < 3000. 2.1.2 Calculate the estimated mean weight-loss of the race car drivers. Step 1: Calculate the midpoint (x_m) for each interval and the product of frequency (f) and midpoint (x_m). The total number of drivers is ∑ f = 40. Interval of Weight Loss w (in grams) — Number of Drivers (f) — Midpoint of Interval (x_m) — f · x_m0 ≤ w < 500 — 1 — (0+500)/(2) = 250 — 1 × 250 = 250; 500 ≤ w < 1000 — 2 — (500+1000)/(2) = 750 — 2 × 750 = 1500; 1000 ≤ w < 1500 — 3 — (1000+1500)/(2) = 1250 — 3 × 1250 = 3750; 1500 ≤ w < 2000 — 8 — (1500+2000)/(2) = 1750 — 8 × 1750 = 14000; 2000 ≤ w < 2500 — 6 — (2000+2500)/(2) = 2250 — 6 × 2250 = 13500; 2500 ≤ w < 3000 — 15 — (2500+3000)/(2) = 2750 — 15 × 2750 = 41250; 3000 ≤ w < 3500 — 5 — (3000+3500)/(2) = 3250 — 5 × 3250 = 16250 Total — 40 — 90500 Step 2: Calculate the estimated mean using the formula x = (∑ (f · x_m))/(∑ f). x = 90500/40 = 2262.5 The estimated mean weight-loss is 2262.5 grams. 2.2.1 Sketch the ogive (cumulative frequency graph) representing race 2 in the ANSWER BOOK. Step 1: Determine the cumulative frequencies for Race 1. Interval of Weight Loss w (in grams) — Number of Drivers (f) — Upper Boundary — Cumulative Frequency (CF)0 ≤ w < 500 — 1 — 500 — 1 500 ≤ w < 1000 — 2 — 1000 — 1+2=3; 1000 ≤ w < 1500 — 3 — 1500 — 3+3=6; 1500 ≤ w < 2000 — 8 — 2000 — 6+8=14; 2000 ≤ w < 2500 — 6 — 2500 — 14+6=20; 2500 ≤ w < 3000 — 15 — 3000 — 20+15=35; 3000 ≤ w < 3500 — 5 — 3500 — 35+5=40 Step 2: Determine the value of k. The problem states that the amount of weight lost in race 2 was k grams more than in race 1. For Race 1, the minimum weight loss is 0 grams. For Race 2, the ogive is grounded at (4; 0), meaning the minimum weight loss is 4 grams. So, 0 + k = 4 ⇒ k = 4. Step 3: Determine the points for the ogive of Race 2 by adding k=4 to the upper boundaries of Race 1. The ogive for Race 2 will be plotted using the following points: Start: (4; 0) (given) (500+4; 1) = (504; 1); (1000+4; 3) = (1004; 3); (1500+4; 6) = (1504; 6); (2000+4; 14) = (2004; 14); (2500+4; 20) = (2504; 20); (3000+4; 35) = (3004; 35); End: (3500+4; 40) = (3504; 40) (given) To sketch the ogive: Plot the points (4; 0), (504; 1), (1004; 3), (1504; 6), (2004; 14), (2504; 20), (3004; 35), and (3504; 40). Connect these points with a smooth curve. The x-axis represents weight loss (in grams) and the y-axis represents cumulative frequency (number of drivers). 2.2.2 How will the range of race 2 compare with the range of race 1? Step 1: Calculate the range for Race 1. The minimum weight loss for Race 1 is 0 grams. The maximum weight loss for Race 1 is the upper boundary of the last interval, which is 3500 grams. Range₁ = Maximum value − Minimum value = 3500 − 0 = 3500 grams Step 2: Calculate the range for Race 2. The minimum weight loss for Race 2 is 4 grams (given as grounded at (4; 0)). The maximum weight loss for Race 2 is 3504 grams (given as the maximum value of the ogive (3504; 40)). Range₂ = Maximum value − Minimum value = 3504 − 4 = 3500 grams Step 3: Compare the ranges. The range of Race 2 is the same as the range of Race 1. 2.2.3 Determine the average weight lost in race 2. Step 1: Recall the estimated mean weight loss for Race 1 from question 2.1.2. Estimated mean for Race 1 (x_1) = 2262.5 grams. Step 2: Use the relationship between the weight loss in Race 1 and Race 2. The amount of weight lost in Race 2 was k=4 grams more than in Race 1. When all data points are shifted by a constant value, the mean also shifts by that same constant value. x_2 = x_1 + k; x_2 = 2262.5 + 4 = 2266.5 The average weight lost in Race 2 is 2266.5 grams.

Interval of Weight Loss $w$ (in grams)	Number of Drivers ( $f$ )	Midpoint of Interval ( $x_m$ )	$f \cdot x_m$
$0 \le w < 500$	1	$\frac{0+500}{2} = 250$	$1 \times 250 = 250$
$500 \le w < 1000$	2	$\frac{500+1000}{2} = 750$	$2 \times 750 = 1500$
$1000 \le w < 1500$	3	$\frac{1000+1500}{2} = 1250$	$3 \times 1250 = 3750$
$1500 \le w < 2000$	8	$\frac{1500+2000}{2} = 1750$	$8 \times 1750 = 14000$
$2000 \le w < 2500$	6	$\frac{2000+2500}{2} = 2250$	$6 \times 2250 = 13500$
$2500 \le w < 3000$	15	$\frac{2500+3000}{2} = 2750$	$15 \times 2750 = 41250$
$3000 \le w < 3500$	5	$\frac{3000+3500}{2} = 3250$	$5 \times 3250 = 16250$
Total	40		90500

Interval of Weight Loss $w$ (in grams)	Number of Drivers ( $f$ )	Upper Boundary	Cumulative Frequency (CF)
$0 \le w < 500$	1	500	1
$500 \le w < 1000$	2	1000	$1+2=3$
$1000 \le w < 1500$	3	1500	$3+3=6$
$1500 \le w < 2000$	8	2000	$6+8=14$
$2000 \le w < 2500$	6	2500	$14+6=20$
$2500 \le w < 3000$	15	3000	$20+15=35$
$3000 \le w < 3500$	5	3500	$35+5=40$

Write down the modal class of the data. Calculate the estimated mean weight-loss of the race car drivers. Sketch the ogive (cumulative frequency graph) representing race 2 in the ANSWER BOOK. How will the range of race 2 compare with the range of race 1? Determine the average weight lost in race 2.

Step-by-step solution

Identify the class interval with the highest frequency.

Calculate the midpoint ( $x_m$ ) for each interval and the product of frequency ( $f$ ) and midpoint ( $x_m$ ).

Calculate the estimated mean using the formula $\bar{x} = \frac{\sum (f \cdot x_m)}{\sum f}$ .

Determine the cumulative frequencies for Race 1.

Determine the value of $k$ .

Determine the points for the ogive of Race 2 by adding $k=4$ to the upper boundaries of Race 1.

Calculate the range for Race 1.

Calculate the range for Race 2.

Compare the ranges.

Recall the estimated mean weight loss for Race 1 from question 2.1.2.

Use the relationship between the weight loss in Race 1 and Race 2.

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More Mathematics Questions

Write down the modal class of the data. Calculate the estimated mean weight-loss of the race car drivers. Sketch the ogive (cumulative frequency graph) representing race 2 in the ANSWER BOOK. How will the range of race 2 compare with the range of race 1? Determine the average weight lost in race 2.

Step-by-step solution

Identify the class interval with the highest frequency.

Calculate the midpoint (xmx_mxm​) for each interval and the product of frequency (fff) and midpoint (xmx_mxm​).

Calculate the estimated mean using the formula xˉ=∑(f⋅xm)∑f\bar{x} = \frac{\sum (f \cdot x_m)}{\sum f}xˉ=∑f∑(f⋅xm​)​.

Determine the cumulative frequencies for Race 1.

Determine the value of kkk.

Determine the points for the ogive of Race 2 by adding k=4k=4k=4 to the upper boundaries of Race 1.

Calculate the range for Race 1.

Calculate the range for Race 2.

Compare the ranges.

Recall the estimated mean weight loss for Race 1 from question 2.1.2.

Use the relationship between the weight loss in Race 1 and Race 2.

Still stuck on this one?

More Mathematics Questions

Calculate the midpoint ( $x_m$ ) for each interval and the product of frequency ( $f$ ) and midpoint ( $x_m$ ).

Calculate the estimated mean using the formula $\bar{x} = \frac{\sum (f \cdot x_m)}{\sum f}$ .

Determine the value of $k$ .

Determine the points for the ogive of Race 2 by adding $k=4$ to the upper boundaries of Race 1.