Question

homework 6.2
name
multiple choice. choose the one alternative that best completes the statement or answers the question.
assume that x has a normal distribution, and find the indicated probability.

1. the mean is μ = 15.2 and the standard deviation is σ = 0.9.

find the probability that x is greater than 15.2.
a) 0.9998 b) 0.0003 c) 1.0000 d) 0.5000

1. the mean is μ = 15.2 and the standard deviation is σ = 0.9.

find the probability that x is greater than 17.
a) 0.0228 b) 0.9821 c) 0.9772 d) 0.9713

1. the mean is μ = 22.0 and the standard deviation is σ = 2.4.

find the probability that x is between 19.7 and 25.3.
a) 0.7477 b) 1.0847 c) 0.4107 d) 0.3370
find the indicated probability.

1. the incomes of trainees at a local mill are normally distributed with a mean of $1100 and a

standard deviation of $150. what percentage of trainees earn less than $900 a month?
a) 9.18% b) 40.82% c) 35.31% d) 90.82%

1. the diameters of pencils produced by a certain machine are normally distributed with a mean of

0.30 inches and a standard deviation of 0.01 inches. what is the probability that the diameter of a
randomly selected pencil will be less than 0.285 inches?
a) 0.9332 b) 0.0596 c) 0.0668 d) 0.4332

1. a banks loan officer rates applicants for credit. the ratings are normally distributed with a

mean of 200 and a standard deviation of 50. if an applicant is randomly selected, find the
probability of a rating that is between 200 and 275.
a) 0.0668 b) 0.4332 c) 0.9332 d) 0.5

1. in one region, the september energy consumption levels for single - family homes are found to be

normally distributed with a mean of 1050 kwh and a standard deviation of 218 kwh. for a
randomly selected home, find the probability that the september energy consumption level is
between 1100 kwh and 1225 kwh.
a) 0.2881 b) 0.3791 c) 0.1971 d) 0.0910

1. assume that the weights of quarters are normally distributed with a mean of 5.67 g and a

standard deviation 0.070 g. a vending machine will only accept coins weighing between 5.48 g
and 5.82 g. what percentage of legal quarters will be rejected?
a) 1.62% b) 0.0196% c) 1.96% d) 2.48%

Explanation:

Step1: Recall property of normal distribution

In a normal - distribution $N(\mu,\sigma)$, the probability density function is symmetric about the mean $\mu$. So $P(X > \mu)=0.5$.
For question 1, since $\mu = 15.2$, then $P(X>15.2) = 0.5$.

Step2: Calculate z - score

The z - score is calculated as $z=\frac{x-\mu}{\sigma}$. For question 2, $\mu = 15.2$, $\sigma = 0.9$, and $x = 17$. Then $z=\frac{17 - 15.2}{0.9}=\frac{1.8}{0.9}=2$. Using the standard normal table, $P(X>17)=1 - P(Z\leq2)=1 - 0.9772 = 0.0228$.

Step3: Calculate z - scores for both values

For question 3, when $\mu = 22.0$, $\sigma = 2.4$, for $x_1 = 19.7$, $z_1=\frac{19.7 - 22.0}{2.4}=\frac{- 2.3}{2.4}\approx - 0.96$. For $x_2 = 25.3$, $z_2=\frac{25.3 - 22.0}{2.4}=\frac{3.3}{2.4}=1.375$. Then $P(19.7

Step4: Calculate z - score

For question 4, $\mu = 1100$, $\sigma = 150$, and $x = 900$. Then $z=\frac{900 - 1100}{150}=\frac{-200}{150}\approx - 1.33$. Using the standard - normal table, $P(X < 900)=P(Z < - 1.33)=1 - 0.9082 = 0.0918 = 9.18\%$.

Step5: Calculate z - score

For question 5, $\mu = 0.30$, $\sigma = 0.01$, and $x = 0.285$. Then $z=\frac{0.285 - 0.30}{0.01}=\frac{-0.015}{0.01}=-1.5$. Using the standard - normal table, $P(X < 0.285)=P(Z < - 1.5)=1 - 0.9332 = 0.0668$.

Step6: Calculate z - score

For question 6, $\mu = 200$, $\sigma = 50$, for $x_1 = 200$, $z_1 = 0$, for $x_2 = 275$, $z_2=\frac{275 - 200}{50}=1.5$. Then $P(200

Step7: Calculate z - scores

For question 7, $\mu = 1050$, $\sigma = 218$, for $x_1 = 1100$, $z_1=\frac{1100 - 1050}{218}\approx0.23$, for $x_2 = 1225$, $z_2=\frac{1225 - 1050}{218}\approx0.8$. Then $P(1100

Step8: Calculate z - scores

For question 8, $\mu = 5.67$, $\sigma = 0.070$, for $x_1 = 5.48$, $z_1=\frac{5.48 - 5.67}{0.070}=\frac{-0.19}{0.070}\approx - 2.71$, for $x_2 = 5.82$, $z_2=\frac{5.82 - 5.67}{0.070}=\frac{0.15}{0.070}\approx2.14$. The probability of acceptance is $P(5.48

Answer:

1. D. 0.5000
2. A. 0.0228
3. A. 0.7477
4. A. 9.18%
5. C. 0.0668
6. B. 0.4332
7. C. 0.1971
8. C. 1.96%