The mean finish time for a yearly amateur auto race was 186.37 minutes with a standard deviation of 0.323 minute. the winning car, driven by ted, finished in 185.82 minutes. the previous year's race had a mean finishing time of 112.6 with a standard deviation of 0.141 minute. the winning car that year, driven by karen, finished in 112.25 minutes. find their respective z-scores. who had the more convincing victory?

Ted had a more convincing victory with about 4.46%

The z score is used to determine by how many standard deviations the raw score is above or below the mean.

The z score is given by:

For ted, μ = 186.37 minutes , σ = 0.323

For x < 185.82 minutes:

From the normal distribution table, P(z < -1.70) = 0.0446 = 4.46%

For karen, μ = 112.6 minutes , σ = 0.141

For x < 112.25 minutes:

From the normal distribution table, P(z < -2.48) = 0.0066 = 0.6%

Hence Ted had a more convincing victory with about 4.46%

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step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt