AP Stats - Test #8

AP Statistics

Unit Test #8

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The fuel economy for a car varies approximately normally with mean 24.8 mpg and standard deviation 6.2 mpg. Use this information for all questions on this page.

Calculate and interpret the standardized score of a car that has a fuel economy of 19 mpg.

(5)

$\frac{19 - 24.8}{6.2} = - 0.9355$ ; this car’s fuel economy is 0.9355 standard deviations below the mean (which is not unusual).

According to the Empirical Rule, what approximate percentage of cars have fuel economies of 18.6 mpg or less?

(3)

Since 18.6 is one standard deviation below the mean, about 16% of cars will have fuel economies of 18.6 mpg or less.

According to the Empirical Rule, approximately 2.5% of cars have fuel economies that are greater than what amount?

(3)

The upper 2.5% of fuel economies will have standardized scores of 2 or more; that translates to fuel economies of $(2) (6.2) + 24.8 = 37.2$ mpg or more.

Find the probability that a randomly selected car will have a fuel economy between 18 mpg and 25 mpg.

(5)

$P (18 < x < 25) = P (\frac{18 - 24.8}{6.2} < z < \frac{25 - 24.8}{6.2}) = P (- 1.0968 < z < 0.0323) \approx 0.3765$

The 1% of cars with the worst fuel economies get how many miles to the gallon?

(5)

The lowest 1% will have z-scores of $- 2.3263$ and lower. That translates to fuel economies of $x = (- 2.3263) (6.2) + 24.8 = 10.3766$ mpg or lower.

An airline has found that weight of a piece of luggage brought by a passenger (X) varies with mean 40 pounds and standard deviation 8 pounds. A particular plane can carry 200 passengers—assume that each passenger brings one piece of luggage.

Use this information for all questions on this page.

Find $μ_{\bar{x}}$ .

(2)

$μ_{\bar{x}} = μ = 40$

Find $σ_{\bar{x}}$ .

(4)

$σ_{\bar{x}} = \frac{σ}{\sqrt{n}} = \frac{8}{\sqrt{200}} = 0.5657$

Describe the shape of the sampling distribution of $\bar{x}$ . Justify your answer.

(4)

Since the size of the sample is quite large (well over 30), the shape of the sampling distribution of $\bar{x}$ will be approximately normal.

Find the probability that the mean weight of the luggage is greater than 41 pounds.

(5)

$P (\bar{x} > 41) = P (z > \frac{41 - 40}{0.5657}) = P (z > 1.7678) \approx 0.0386$

10.

Find the probability that the mean weight of the luggage is less than 39.5 pounds.

(5)

$P (\bar{x} < 39.5) = P (z < \frac{39.5 - 40}{0.5657}) = P (z < - 0.8839) \approx 0.1884$

A large corporation has found, over the years, that about 10% of its sales trainees are rated as outstanding at the completion of their training program. A particular office of this corporation had 150 sales trainees in a recent year. Let $\hat{p}$ be the proportion of sales trainees at this location that will be rated as outstanding.

Use this information for all questions on this page.

11.

Find $μ_{\hat{p}}$ .

(2)

$μ_{\hat{p}} = p = 0.1$

12.

Find $σ_{\hat{p}}$ .

(5)

$σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}} = \sqrt{\frac{(0.1) (0.9)}{150}} \approx 0.0245$

13.

Describe the shape of the sampling distribution of $\hat{p}$ . Justify your answer.

(5)

The shape will be approximately normal because $n p = (150) (0.1) = 15 > 10$ and $n (1 - p) = (150) (0.9) = 135 > 10$ .

14.

Find the probability that at least 17 of the trainees will be rated as outstanding.

(5)

$P (X \geq 17) = P (\hat{p} > \frac{17}{150}) = P (\hat{p} > 0.1133) = P (z > \frac{0.1133 - 0.1}{0.0245}) = P (z > 0.5443) \approx 0.2931$

15.

Find the probability that fewer than 14 of the trainees will be rated as outstanding.

(5)

$P (X < 14) = P (\hat{p} < \frac{14}{150}) = P (\hat{p} < 0.0933) = P (z < \frac{0.0933 - 0.1}{0.0245}) = P (z < - 0.2722) \approx 0.3927$

Page last updated 10:26 2020-01-30