Question

listed below are head injury measurements from small cars that were tested in crashes. the measurements are in hic, which is a measurement of a standard head injury criterion. (lower hic values correspond to safer cars). the listed values correspond to cars a, b, c, d, e, f, and g, respectively. find the a. mean, b. median, c. midrange, and d. mode for the data. also complete parts e. and f.
a. find the mode.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the mode(s) is (are) . (use a comma to separate answers as needed.)
b. there is no mode.

e. which car appears to be the safest?
car b
car d
car g
car a
car c
car e
car f

f. based on these limited results, do small cars appear to have about the same risk of head injury in a crash?
a. yes, because the data values differ substantially.
b. no, because the data values differ substantially.
c. yes, because the data values do not differ substantially.
d. no, because the data values do not differ substantially.

Explanation:

Part a: Find the mode

Step 1: Recall the definition of mode

The mode is the value that appears most frequently in a data set. If all values appear with the same frequency, there is no mode.

Step 2: Analyze the given data

The data values are: 350, 301, 568, 397, 344, 513, 470. Let's count the frequency of each value:

• 350: appears once
• 301: appears once
• 568: appears once
• 397: appears once
• 344: appears once
• 513: appears once
• 470: appears once

Since each value appears exactly once, there is no mode.

Step 1: Identify the lowest "hic" value

First, we find the minimum value in the data set: 301, 344, 350, 397, 470, 513, 568. The lowest value is 301.

Step 2: Determine the corresponding car

We need to know which car corresponds to the value 301. (Assuming the data is ordered as cars A - G with values 350 (A?), 301 (B?), 568 (C?), 397 (D?), 344 (E?), 513 (F?), 470 (G?) - from the options in part e, the car with the lowest value (safest) would be the one with 301, which is Car B (from the options: car B is an option, and assuming 301 is Car B's value).

Step 1: Analyze the data spread

The data values are 301, 344, 350, 397, 470, 513, 568. Let's check the range: maximum - minimum = 568 - 301 = 267. Also, the values are spread out (301 to 568), but wait, actually, let's check the options. The options are about whether the data values differ substantially. Wait, maybe I misread. Wait, the options: A. Yes, because data differ substantially (but that would mean different risk), B. No, because data differ substantially (contradictory), C. Yes, because data do not differ substantially, D. No, because data do not differ substantially. Wait, no - the question is "do small cars appear to have about the same risk" - same risk would mean data values are close (not differ substantially). But the data values are 301, 344, 350, 397, 470, 513, 568. Let's check the differences: 344 - 301 = 43, 350 - 344 = 6, 397 - 350 = 47, 470 - 397 = 73, 513 - 470 = 43, 568 - 513 = 55. The values are spread, but maybe the key is that there's no single mode (from part a) and the values are not all the same, but do they differ substantially? Wait, the options: D says "No, because the data values do not differ substantially" - no, that's contradictory. Wait, maybe the correct approach is: if the data values are close (not differ much), then same risk. But the values range from 301 to 568, which is a large range. Wait, maybe I made a mistake. Wait, the options: Let's re - read the question: "do small cars appear to have about the same risk of head injury in a crash?"

If the data values (hic) are close to each other, then the risk is about the same. If they differ a lot, then the risk is different. Let's calculate the range: 568 - 301 = 267. The mean (from part a, if we calculate it: (301 + 344 + 350 + 397 + 470 + 513 + 568)/7=(301+344=645; 645+350=995; 995+397=1392; 1392+470=1862; 1862+513=2375; 2375+568=2943)/7≈420.43. The standard deviation (approximate): sum of squared differences from mean:

(301 - 420.43)²≈(-119.43)²≈14263.52

(344 - 420.43)²≈(-76.43)²≈5841.54

(350 - 420.43)²≈(-70.43)²≈4960.38

(397 - 420.43)²≈(-23.43)²≈549.06

(470 - 420.43)²≈(49.57)²≈2457.18

(513 - 420.43)²≈(92.57)²≈8569.20

(568 - 420.43)²≈(147.57)²≈21777.90

Sum of these: 14263.52+5841.54 = 20105.06; +4960.38 = 25065.44; +549.06 = 25614.5; +2457.18 = 28071.68; +8569.20 = 36640.88; +21777.90 = 58418.78

Variance = 58418.78/7≈8345.54, standard deviation≈√8345.54≈91.35. The coefficient of variation (standard deviation/mean)≈91.35/420.43≈0.217, which is about 21.7%, which is a moderate variation. But the options: Let's look at the options again. Option D: "No, because the data values do not differ substantially" - no, that's wrong. Option B: "No, because the data values differ substantially" - if the data differs substantially, then the risk is different, so the answer would be B? Wait, no - the question is "do small cars appear to have about the same risk" - same risk would mean data not differ substantially. But the data does differ (range 267, standard deviation ~91). Wait, maybe the intended answer is D? No, that doesn't make sense. Wait, maybe I misread the data. Wait, the data given in the problem: 350, 301, 568, 397, 344, 513, 470. Let's sort them: 301, 344, 350, 397, 470, 513, 568. The differences between consecutive values: 344 - 301 = 43, 350 - 344 = 6, 397 - 350 = 47, 470 - 397 = 73, 513 - 470 = 43, 568 - 513 = 55. Some differences are small (6), some are moderate (43,47,55), some are larger (73). But the options: A. Yes, because data differ substantially (but same risk would mean not differ), B. No, because data differ substanti…

Answer:

There is no mode.

Part e: Which car appears to be the safest?

(Assuming lower "hic" values mean safer, as lower values correspond to safer cars from the context)