A certain brand of automobile tire has a mean life span of 32,000 miles and a standard deviation of 2,450 miles. (Assume the life spans of the tires have a bell-shaped distribution.) (a) The life spans of three randomly selected tires are 34,000 miles, 37,000 miles, and 31,000 miles. Find the z-score that corresponds to each life span.

a)

The z-score corresponds to 34000 is 0.816

The z-score corresponds to 37000 is 2.041

The z-score corresponds to 31000 is -0.408

Step-by-step explanation:

The z-score corresponds to 34000

Let x=34000

z-score=(x-mean)/standard deviation

z-score=34000-32000/2450=0.816

The z-score corresponds to 34000 is 0.816

The z-score corresponds to 37000

Let x=37000

z-score=(x-mean)/standard deviation

z-score=37000-32000/2450=2.041

The z-score corresponds to 37000 is 2.041

The z-score corresponds to 31000

Let x=31000

z-score=(x-mean)/standard deviation

z-score=31000-32000/2450=-0.408

The z-score corresponds to 31000 is -0.408

Using slope-intercept form, y = mx + b where m = slope and b = y-intercept:

We know our slope is -6. This can be interpreted as -6/1, which rise-over-run-wise, means that when y changes by 6, x changes inversely by 1.
To find that y-intercept, though, we need to find the value of y when x = 0.
Use our point (-9, -3) to find this...
We want to add 9 to x so that it becomes 0.
According to our slope, this means subtracting 54 from y.
Our y-intercept is at (0, -57), with -57 being the value of b we put in our equation.



You could also just use point-slope form:
y - y¹ = m(x - x¹)
y - (-3) = -6(x - (-9))
y + 3 = -6(x + 9)
And convert to slope-intercept if you want:
y + 3 = -6x - 54
y = -6x - 57