Normal Probability

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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) $\frac{60 - 75}{10}$
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?
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
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.
5.
Exam scores are normally distributed with $\mu = 72$ and $\sigma = 6$ . The table gives $P(Z < 1.5) = 0.9332$ . Find $P(X < 81)$ .
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.
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
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.
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 $P(156 < X < 171)$ .
10.
IQ scores are normally distributed with $\mu = 100$ and $\sigma = 15$ . Find the 90th percentile.
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
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.
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?
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.
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
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.
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?
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?