Question

Score: 0 of 13 pts 2 of 4 y HW Score: 0%, 0 of 50 pts 7.1.46 Question Help The police department of a major city needs to update its budget. For this purpose, they need to understand the variation in their fines collected from motorists for speeding. As a sample, they recorded the speeds of cars driving past a location with a 25 mph speed limit, a place that in the past has been known for producing fines. The mean of 100 representative readings was 28.76 mph, with a standard deviation of 3.36 mph. a) How many standard deviations from the mean would a car going the speed limit be? b) Which would be more unusual, a car traveling 38 mph or one going 17 mph?

Answer 1

We are given a sample of size 100 with mean $\mu$ = 28.76 mph and a standard deviation of $\sigma$ = 3.36 mph.

a)

To find how many standard deviations from the mean would a car going the speed limit (i.e. 25 mph) be, we find the z-value of the speed limit (25 mph):
$\begin{align*} \text{z-value} &= \frac{25-\mu}{\sigma} \\ &= \frac{25-28.76}{3.36} \\&= \frac{-3.76}{3.36} \\ &= \bf -1.119048 \ \ \ \ \ [ANSWER] \end{align*}$

Thus, a car going the speed limit would be -1.119048 standard deviations from the mean. [ANSWER]

b)

To find what would be more unusual a car travelling at 38 mph or going at 17 mph, we find the z-values of both:

$\begin{align*} \text{z-value(38)} &= \frac{38-\mu}{\sigma} \\ &= \frac{38-28.76}{3.36} \\&= \frac{9.24}{3.36} \\ &= 2.75 \\ \text{z-value(17)} &= \frac{17-\mu}{\sigma} \\ &= \frac{17-28.76}{3.36} \\&= \frac{-11.76}{3.36} \\ &= -3.5 \end{align*}$

Now, a z-value is unusual if it is far from 0. Moreover, to compare two z-values we see which one is farther from 0 than the other.

Here, clearly z-value(17) = -3.5 is farther from zero than z-value(38) = 2.75.

Thus, a car travelling at 17 mph is more unusual. [ANSWER]

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