AP Statistics - NP: Normal Probabilities

1 Probability > Normal probabilities · Level 3

Diet Plan A advertises an average monthly weight loss of 10 pounds with a standard deviation of 3 pounds. Diet Plan B claims an average monthly weight loss of 12 pounds with a standard deviation of 1.8 pounds. Assuming both assertions are correct and assuming roughly normal distributions, which diet plan is more likely to result in a monthly weight loss of over 15 pounds?

A

Diet Plan A is more likely to result in a monthly weight loss of over 15 pounds because of its greater standard deviation.

B

Diet Plan B is more likely to result in a monthly weight loss of over 15 pounds because of its greater mean.

C

For both plans, the probability of a weight loss over 15 pounds is 0.04779.

D

For both plans, the probability of a weight loss over 15 pounds is 0.95221.

E

The problem cannot be solved from the information given.

2 Probability > Normal probabilities · Level 3

Which of the following are true statements? I. The area under a normal curve is always equal to 1, no matter the mean and standard deviation. II. The smaller the standard deviation of a normal curve, the higher and narrower the graph is. III. Normal curves with different means are centered around different numbers.

A

I and II only

B

I and III only

C

II and III only

D

I, II, and III

E

None of the above gives the complete set of true responses.

3 Probability > Normal probabilities · Level 3

A cell phone takes an average of 11 minutes to move through an assembly line. If the standard deviation is 2 minutes and if the distribution is roughly normal, what is the probability that a cell phone will take over 12 minutes to move through the assembly line?

A

$P\left(z > \dfrac{12-11}{2}\right)$

B

$P\left(z > \dfrac{12-11}{\dfrac{2}{\sqrt{2}}}\right)$

C

$P\left(z > \dfrac{11-12}{2}\right)$

D

$P\left(z > \dfrac{11-12}{\dfrac{2}{\sqrt{2}}}\right)$

E

$2P\left(t > \dfrac{11-12}{2}\right)$

4 Probability > Normal probabilities · Level 3

A bowler's scores are approximately normally distributed with a mean of 210. What is the standard deviation if 30 percent of her scores are below 200?

A

$\dfrac{200-210}{-0.5244}$

B

$\dfrac{200-200}{-0.5244}$

C

$\dfrac{200-210}{0.5244}$

D

$\dfrac{200-210}{0.4756}$ 210−200

E

$\dfrac{210-200}{0.4756}$

5 Probability > Normal probabilities · Level 3

Which of the following is a true statement?

A

The area under the standard normal curve between 0 and 2 is twice the area between 0 and 1.

B

The area under the standard normal curve between 0 and 2 is half the area between −2 and 2.

C

For the standard normal curve, the interquartile range is approximately 3.

D

For the standard normal curve, approximately 1 out of 1,000 values are greater than 10.

E

The 68-95-99.7 rule applies only to normal curves where the mean and standard deviation are known.

6 Probability > Normal probabilities · Level 3

The starting national average salary for a computer security specialist is \$58,760. Assuming a roughly normal distribution and a standard deviation of \$6,500, what is the probability that a randomly chosen computer security specialist will start with a salary between \$50,000 and \$60,000?

A

$P\left(\dfrac{50000}{6500} < z < \dfrac{60000}{6500}\right)$

B

$2P\left(z > \dfrac{60000-50000}{6500}\right)$

C

$P\left(\dfrac{58760-50000}{6500} < z < \dfrac{58760-60000}{6500}\right)$

D

$P\left(\dfrac{58760-50000}{\dfrac{6500}{\sqrt{n}}} < z < \dfrac{58760-60000}{\dfrac{6500}{\sqrt{n}}}\right)$

E

$P\left(\dfrac{50000-58760}{6500} < z < \dfrac{60000-58760}{6500}\right)$

7 Probability > Normal probabilities · Level 3

Populations P₁ and P₂ are roughly normally distributed and have identical means. However, the standard deviation of P₁ is half the standard deviation of P₂. What can be said about the percentage of observations falling within one standard deviation of the mean for each population?

A

The percentage for P₁ is twice the percentage for P₂.

B

The percentage for P₁ is greater, but not twice as great, as the percentage for P₂.

C

The percentage for P₂ is twice the percentage for P₁.

D

The percentage for P₂ is greater, but not twice as great, as the percentage for P₁.

E

The percentages are identical.

8 Probability > Normal probabilities · Level 3

A college library determines that books are checked out for an average of 8 days with a standard deviation of 2 days. Assuming a roughly normal distribution, what is the shortest time interval for which two-thirds of the books are out?

A

0 to 8.9 days

B

6.1 to 7.1 days

C

6.1 to 8 days

D

6.1 to 9.9 days

E

7.1 to 14 days

9 Probability > Normal probabilities · Level 3

A set of employee commuting distances from work has a roughly normal distribution with a mean of 12 miles and a standard deviation of 3.5 miles. If a randomly selected employee commutes over 10 miles, what is the probability she commutes under 15 miles?

A

0.521

B

0.545

C

0.647

D

0.716

E

0.727

10 Probability > Normal probabilities · Level 3

Which of the following statements is incorrect?

A

In all normal distributions, the mean and median are equal.

B

Bell-shaped curves may not have normal distributions.

C

Virtually all the area under a normal curve is within three standard deviations of the mean, no matter the particular mean and standard deviation.

D

A normal distribution is completely determined by two numbers, its mean and its standard deviation.

E

Standardized scores (z-scores) always have a normal distribution no matter the original distribution.

11 Probability > Normal probabilities · Level 3

A trucking firm determines that the miles per gallon (mpg) achieved by trucks in its fleet are roughly normally distributed with a standard deviation of 2.5 mpg. What is the mean mpg if 75 percent of the trucks achieve better than 13.2 mpg?

A

13.2 + 0.3255(2.5)

B

13.2 + 0.6745(2.5)

C

13.2 + 0.7500(2.5)

D

13.2 − 0.6745(2.5)

E

13.2 − 0.7500(2.5)

12 Probability > Normal probabilities · Level 3

The mean yearly medical expenses (including insurance payments) for individuals in a large city is \$15,300 with a standard deviation of \$3,600. Assuming a roughly normal distribution, what is the probability that two randomly chosen individuals in the city both have yearly medical expenses over \$20,000?

A

$2P\left(z > \dfrac{20000-15300}{3600}\right)$

B

$[P\left(z > \dfrac{20000-15300}{3600}\right)]^2$

C

$\dfrac{1}{2}P\left(z > \dfrac{20000-15300}{3600}\right)$

D

$P\left(z > \dfrac{20000-15300}{\sqrt{3600}}\right)$

E

$P\left(z > \dfrac{20000-15300}{\dfrac{3600}{\sqrt{2}}}\right)$

13 Probability > Normal probabilities · Level 3

The monthly rental paid per person for college students living off campus has a roughly normal distribution with a mean of \$275 and a standard deviation of \$40. Ninety percent of the rentals are greater than what amount?

A

$275 − 1.282($40)

B

$275 − 1.645($40)

C

$275 − 1.96($40)

D

$275 + 1.282($40)

E

$275 + 1.96($40)

14 Probability > Normal probabilities · Level 3

Suppose women's foot lengths are roughly normally distributed with a mean of 20 cm and a standard deviation of 4 cm, while men's foot lengths are roughly normally distributed with a mean of 26 cm and a standard deviation of 5 cm. If a policeman measures a footprint to be 22.5 cm, is there a greater probability that it belongs to a man or to a woman?

A

Man, because 22.5 − 20 < 26 − 22.5

B

Man, because $\dfrac{2.5}{4} < \dfrac{3.5}{5}$

C

Woman, because 22.5 − 20 < 26 − 22.5

D

Woman, because $\dfrac{2.5}{4} < \dfrac{3.5}{5}$

E

This cannot be answered without knowing if we can assume independence.

15 Probability > Normal probabilities · Level 3

A couple is looking to purchase their first house. In the neighborhood in which they are interested, home prices are roughly normally distributed with a mean of \$275,000 and a standard deviation of \$35,000. They ask the realtor to show them only homes under \$300,000. What percentage of the homes shown to them will be over \$250,000?

A

0.312

B

0.475

C

0.525

D

0.688

E

0.763

16 Probability > Normal probabilities · Level 3

Among 2,500 subjects signing up for a program to stop smoking, the distribution of their daily number of cigarettes was roughly normal with a mean of 27.1 and a standard deviation of 4.6. Which expression below represents the 60th percentile of the distribution?

A

27.1 − (0.25)(4.6)

B

27.1 − (0.40)(4.6)

C

27.1 − (0.60)(4.6)

D

27.1 + (0.25)(4.6)

E

27.1 + (0.40)(4.6)

17 Probability > Normal probabilities · Level 3

The women's 500 m skating times are roughly normally distributed with a mean of 40.44 seconds. The z-score of China's Peiyu Jin's time of 38.69 seconds is −1.944. What percent of all women's 500 m skating times are above 42 seconds?

A

1.56

B

1.75

C

4.15

D

98.25

E

95.85

18 Probability > Normal probabilities · Level 3

The number of miles that a highway construction team can lay on good-weather days is roughly normally distributed with a mean of 3 and a standard deviation of 0.35. The number of miles that a highway construction team can lay on bad-weather days is roughly normally distributed with a mean of 1.9 and a standard deviation of 0.4. If the probability of good weather is 0.7 (and of bad weather is 0.3), what is the probability of laying at least 2.5 miles on a random day?

A

0.06681

B

0.49511

C

0.64638

D

0.6664

E

0.92344

19 Probability > Normal probabilities · Level 3

A distribution of scores is approximately normal with a mean of 28 and a standard deviation of 3.4. Which of the following equations should be used to find a score $x$ with 20 percent of the scores above it?

A

$\dfrac{x-28}{3.4} = 0.80$

B

$\dfrac{x-28}{\sqrt{3.4}} = 0.80$

C

$\dfrac{x-28}{3.4} = 0.84$

D

$\dfrac{x-28}{\sqrt{3.4}} = 0.84$

E

$\dfrac{x-28}{3.4^2} = 0.84$

20 Probability > Normal probabilities · Level 3

A large data set is approximately normally distributed. What is the proper order, from smallest to largest, of $l$, $m$, and $n$, where $l$ = the value with a z-score of –0.6 $m$ = the value of the first quartile $n$ = the value of the 30th percentile

A

$l < m < n$

B

$m < l < n$

C

$m < n < l$

D

$l < n < m$

E

$n < l < m$

21 Probability > Normal probabilities · Level 3

A city's average winter low temperature is 28°F with a standard deviation of 10°F, while the city's average summer low temperature is 66°F with a standard deviation of 6°F. In which season is it more unusual to have a day with a low temperature of 51°F?

A

Winter, because 51 – 28 = 23 is greater than 66 – 51 = 15

B

Winter, because 10 is greater than 6

C

Winter, because 51 sounds like a more unusual temperature for winter than summer

D

Summer, because $|\dfrac{51-66}{6}|$ is more than $|\dfrac{51-28}{10}|$

E

Summer, because $\dfrac{28+66}{2} = 47 < 51$

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