Normal Probability

Show your work for each problem.

1.
Test scores are normally distributed with $\mu = 60$ and $\sigma = 10$ . Which expression gives the z-score of a score of 75?
1. a) $\frac{75 - 60}{10^2}$
2. b) $\frac{75 - 10}{60}$
3. c) $\frac{75 - 60}{10}$
4. d) $\frac{60 - 75}{10}$
Answer: $\frac{75 - 60}{10}$
$z = \frac{x - \mu}{\sigma} = \frac{75 - 60}{10} = 1.5$ . The other expressions swap the order, mix up the numbers or divide by the variance.
2.
Heights of adult men are normally distributed with $\mu = 170$ cm and $\sigma = 8$ cm. What is the z-score of a man who is 182 cm tall?
Answer: 1.5
$z = \frac{182 - 170}{8} = \frac{12}{8} = 1.5$ : he is 1.5 standard deviations above the mean.
3.
The z-table gives $P(Z < 1) = 0.8413$ . What is $P(Z > 1)$ ?
1. a)0.1587
2. b)0.8413
3. c)0.3413
4. d)0.6587
Answer: 0.1587
$P(Z > 1) = 1 - 0.8413 = 0.1587$ . The value 0.8413 is the left area, and 0.3413 is the area between 0 and 1.
4.
Apples from an orchard weigh $\mu = 150$ g with $\sigma = 20$ g, normally distributed. Use $P(Z < 0.5) = 0.6915$ . What proportion of apples weigh less than 160 g? Give a decimal.
Answer: 0.6915
$z = \frac{160 - 150}{20} = 0.5$ . A left area is read straight from the table: $P(X < 160) = P(Z < 0.5) = 0.6915$ , about 69.15% of apples.
5.
Exam scores are normally distributed with $\mu = 72$ and $\sigma = 6$ . The table gives $P(Z < 1.5) = 0.9332$ . Find $P(X < 81)$ .
Answer: 0.9332
1. What is $81 - 72$ ? 9
2. Divide by $\sigma = 6$ . What is the z-score? 1.5
3. What is $P(X < 81)$ as a decimal? 0.9332
4. What percentage of students score below 81? 93.32
6.
Pizza delivery times are normally distributed with $\mu = 30$ min and $\sigma = 5$ min. The table gives $P(Z < 2) = 0.9772$ . Find the probability that a delivery takes more than 40 minutes.
Answer: 0.0228
1. What is the z-score of 40 minutes? 2
2. What is the left area $P(Z < 2)$ ? 0.9772
3. What is $P(X > 40)$ ? 0.0228
7.
Bottles are filled with $\mu = 500$ mL and $\sigma = 5$ mL, normally distributed. Using $P(Z < 2) = 0.9772$ and $P(Z < -1) = 0.1587$ , what is $P(495 < X < 510)$ ?
1. a)0.8185
2. b)1.1359
3. c)0.9772
4. d)0.1587
Answer: 0.8185
$P(495 < X < 510) = P(Z < 2) - P(Z < -1) = 0.9772 - 0.1587 = 0.8185$ . Adding the two values gives more than 1, which is impossible for a probability.
8.
A laptop battery lasts $\mu = 8$ hours with $\sigma = 0.8$ hours, normally distributed. Use $P(Z < 1.25) = 0.8944$ . What is the probability that a battery lasts more than 9 hours? Give a decimal.
Answer: 0.1056
$z = \frac{9 - 8}{0.8} = 1.25$ , so $P(X > 9) = 1 - 0.8944 = 0.1056$ , about 10.56% of batteries.
9.
Women's heights are normally distributed with $\mu = 165$ cm and $\sigma = 6$ cm. The table gives $P(Z < 1) = 0.8413$ and $P(Z < -1.5) = 0.0668$ . Find $P(156 < X < 171)$ .
Answer: 0.7745
1. What is the z-score of 156 cm? -1.5
2. What is the z-score of 171 cm? 1
3. What is $P(156 < X < 171)$ ? 0.7745
10.
IQ scores are normally distributed with $\mu = 100$ and $\sigma = 15$ . Find the 90th percentile.
Answer: 119.2
1. Which z-score has 0.90 of the area to its left? 1.28
2. What is $1.28 \times 15$ ? 19.2
3. What is the 90th percentile $x = 100 + 19.2$ ? 119.2
11.
For women's heights ( $\mu = 165$ cm, $\sigma = 6$ cm) we found $P(X < 174) = 0.9332$ . What does this mean?
1. a)About 93.32% of women are exactly 174 cm tall
2. b)About 93.32% of women are shorter than 174 cm
3. c)174 cm is 93.32% of the mean height
4. d)About 93.32% of women are taller than 174 cm
Answer: About 93.32% of women are shorter than 174 cm
A probability under the normal curve is a proportion of the population: 93.32% of women are shorter than 174 cm, so only 6.68% are taller.
12.
Exam scores are normally distributed with $\mu = 70$ and $\sigma = 8$ . The table gives $P(Z < 0.5) = 0.6915$ . What proportion of students score between 66 and 74? Give a decimal.
Answer: 0.383
$z = \pm 0.5$ . $P(Z < -0.5) = 1 - 0.6915 = 0.3085$ , so $P(66 < X < 74) = 0.6915 - 0.3085 = 0.383$ , about 38.3% of students.
13.
A factory makes 1000 light bulbs. Their lifetimes are normally distributed with $\mu = 1200$ hours and $\sigma = 100$ hours. The table gives $P(Z < 2) = 0.9772$ . About how many bulbs last more than 1400 hours?
Answer: 23
1. What is the z-score of 1400 hours? 2
2. What is $P(X > 1400)$ ? 0.0228
3. What is $0.0228 \times 1000$ ? 22.8
4. Round to a whole number of bulbs. 23
14.
A machine fills bottles with $\mu = 500$ mL and $\sigma = 4$ mL. The lowest 5% of bottles are rejected. Find the cut-off volume below which a bottle is rejected.
Answer: 493.42
1. Which z-score has 0.05 of the area to its left? -1.645
2. What is $-1.645 \times 4$ ? -6.58
3. What is the cut-off $x = 500 + (-6.58)$ in mL? 493.42
15.
Men's heights are normally distributed with $\mu = 175$ cm and $\sigma = 7$ cm. Using $z \approx 1.28$ for the 90th percentile, what height marks the tallest 10%?
1. a)166.04 cm
2. b)175.7 cm
3. c)182 cm
4. d)183.96 cm
Answer: 183.96 cm
$x = 175 + 1.28 \times 7 = 175 + 8.96 = 183.96$ cm. 166.04 cm is the 10th percentile (wrong tail), 182 cm is only $\mu + \sigma$ , and 175.7 cm treats 10% as $0.1\sigma$ .
16.
A phone battery lasts $\mu = 20$ hours with $\sigma = 2$ hours, normally distributed. The table gives $P(Z < -0.67) = 0.2514$ . At least how many hours do 75% of batteries last? Give the answer to two decimal places.
Answer: 18.66
75% above the value means 0.25 below it, so $z \approx -0.67$ (table value 0.2514). $x = 20 + (-0.67) \times 2 = 20 - 1.34 = 18.66$ hours.
17.
A warehouse ships 4000 packages a day. Their weights are normally distributed with $\mu = 250$ g and $\sigma = 10$ g. The table gives $P(Z < 1.5) = 0.9332$ and $P(Z < -1) = 0.1587$ . About how many packages weigh between 240 g and 265 g?
Answer: 3098
1. What is the z-score of 240 g? -1
2. What is the z-score of 265 g? 1.5
3. What is $P(240 < X < 265)$ ? 0.7745
4. How many of the 4000 packages is that? 3098
18.
A cereal company fills boxes with $\sigma = 6$ g. The label says 500 g, and only 2.5% of boxes may contain less. What mean $\mu$ should the machine be set to?
Answer: 511.76
1. Which z-score has 0.025 of the area to its left? -1.96
2. How many grams is $1.96\sigma$ , that is $1.96 \times 6$ ? 11.76
3. 500 g lies 11.76 g below the mean. What is $\mu$ ? 511.76