Math  /  Data & Statistics

QuestionSuppose the lengths of sport-utility vehicles (SUV) are Normally distributed with mean μ=190\mu=190 inches and standard deviation σ=5\sigma=5 inches. Marshall just bought a brand-new SUV that is 194.5 inches long and he is interested in knowing what percentage of SUVs is longer than his. Using his statistical knowledge, he drew a normal curve and labeled the appropriate area of interest. Because his value is above the mean, it can be said that his zz-score will be \qquad Plot B zero negative 194.5 positive

Studdy Solution

STEP 1

1. The lengths of SUVs are Normally distributed with a mean μ=190\mu = 190 inches.
2. The standard deviation of the lengths is σ=5\sigma = 5 inches.
3. Marshall's SUV is 194.5 inches long.
4. The zz-score formula for a given value XX in a normal distribution is z=Xμσz = \frac{X - \mu}{\sigma}.
5. The normal distribution curve is symmetric around the mean.

STEP 2

1. Calculate the zz-score for Marshall's SUV.
2. Interpret the sign of the zz-score.
3. Determine the percentage of SUVs longer than Marshall's SUV.

STEP 3

Calculate the zz-score for Marshall's SUV using the formula z=Xμσz = \frac{X - \mu}{\sigma}.
Given X=194.5X = 194.5 inches, μ=190\mu = 190 inches, and σ=5\sigma = 5 inches:
z=194.51905 z = \frac{194.5 - 190}{5}

STEP 4

Simplify the expression to find the zz-score.
z=4.55=0.9 z = \frac{4.5}{5} = 0.9

STEP 5

Interpret the sign of the zz-score. Since z=0.9z = 0.9 is greater than zero, it is:
positive \text{positive}

STEP 6

To determine the percentage of SUVs longer than Marshall's SUV, we need to find the area to the right of z=0.9z = 0.9 on the standard normal distribution curve.
We use standard normal distribution tables or a calculator to find:
P(Z>0.9) P(Z > 0.9)

STEP 7

Using standard normal distribution tables or a calculator, we find that the area to the right of z=0.9z = 0.9 is approximately:
P(Z>0.9)0.1841 P(Z > 0.9) \approx 0.1841
This means approximately 18.41% of SUVs are longer than Marshall's SUV.
Conclusion: The zz-score for Marshall's SUV is positive.

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