Standard Normal Distribution

Show your work for each problem.

1.
Which mean and standard deviation does the standard normal distribution $N(0, 1)$ have?
1. a) $\mu = 1$ , $\sigma = 0$
2. b) $\mu = 0$ , $\sigma = 1$
3. c) $\mu = 1$ , $\sigma = 1$
4. d) $\mu = 0$ , $\sigma = 0$
Answer: $\mu = 0$ , $\sigma = 1$
The standard normal distribution is centered at $\mu = 0$ and has spread $\sigma = 1$ . That is why a value of $Z$ is directly a number of standard deviations from the mean.
2.
The z-table gives $\Phi(0.5) = 0.6915$ . Find $P(Z > 0.5)$ .
Answer: 0.3085
$P(Z > 0.5) = 1 - \Phi(0.5) = 1 - 0.6915 = 0.3085$ .
3.
Read $P(Z \le 1.28)$ from the z-table. Part of the table: row $1.2$ has $0.8980$ in column $0.07$ , $0.8997$ in column $0.08$ and $0.9015$ in column $0.09$ .
Answer: 0.8997
1. Which row do you use (z to one decimal)? 1.2
2. Which column do you use (the second decimal)? 0.08
3. What value do you read? 0.8997
4.
The z-table shows $0.8413$ for $z = 1$ . What does this number mean?
1. a)The height of the curve at $z = 1$
2. b)The area to the left of $z = 1$
3. c)The area between $-1$ and $1$
4. d)The area to the right of $z = 1$
Answer: The area to the left of $z = 1$
The table gives the cumulative area to the left: $\Phi(1) = P(Z \le 1) = 0.8413$ . The area to the right of $z = 1$ is $1 - 0.8413 = 0.1587$ .
5.
Find $P(Z < 0)$ for the standard normal variable $Z$ . Give a decimal.
Answer: 0.5
The curve is symmetric around the mean 0, so exactly half of the area lies to the left: $\Phi(0) = 0.5$ .
6.
Find $P(Z > 2)$ . The z-table gives $\Phi(2) = 0.9772$ .
Answer: 0.0228
1. What is the area to the left of $z = 2$ ? 0.9772
2. What is $1 - 0.9772$ ? 0.0228
7.
The table gives $\Phi(1.2) = 0.8849$ . Which expression equals $P(Z < -1.2)$ ?
1. a) $-0.8849$
2. b) $1 - 0.8849$
3. c) $0.8849 - 0.5$
4. d) $0.8849$
Answer: $1 - 0.8849$
By symmetry $\Phi(-1.2) = 1 - \Phi(1.2) = 1 - 0.8849 = 0.1151$ . An area cannot be negative, and $0.8849$ would be more than half of the curve to the left of a negative $z$ .
8.
Find $P(Z < -0.25)$ using symmetry. The table gives $\Phi(0.25) = 0.5987$ .
Answer: 0.4013
1. By symmetry $P(Z < -0.25) = P(Z > 0.25)$ . What is $\Phi(0.25)$ ? 0.5987
2. What is $1 - 0.5987$ ? 0.4013
9.
The table gives $\Phi(1.75) = 0.9599$ . Find $P(Z < -1.75)$ .
Answer: 0.0401
$\Phi(-1.75) = 1 - \Phi(1.75) = 1 - 0.9599 = 0.0401$ .
10.
Find $P(0.5 < Z < 1.5)$ . Table: $\Phi(0.5) = 0.6915$ , $\Phi(1.5) = 0.9332$ .
Answer: 0.2417
1. What is the left area up to the upper bound, $\Phi(1.5)$ ? 0.9332
2. What is the left area up to the lower bound, $\Phi(0.5)$ ? 0.6915
3. What is $0.9332 - 0.6915$ ? 0.2417
11.
A classmate computed $P(-2 < Z < 2)$ with the z-table. Which result passes the 68-95-99.7 check?
1. a) $0.4772$
2. b) $0.9974$
3. c) $0.9544$
4. d) $0.6826$
Answer: $0.9544$
Within $\pm 2$ standard deviations lies about 95%. The table gives $\Phi(2) - \Phi(-2) = 0.9772 - 0.0228 = 0.9544$ . $0.6826$ belongs to $\pm 1$ , $0.9974$ to $\pm 3$ , and $0.4772$ is only the half from 0 to 2.
12.
Find $P(-1 < Z < 1)$ with the table value $\Phi(1) = 0.8413$ , then compare with the 68-95-99.7 rule.
Answer: 0.6826
1. What is $\Phi(-1) = 1 - 0.8413$ ? 0.1587
2. What is $0.8413 - 0.1587$ ? 0.6826
3. Round your result to a whole percent. What percent is it? 68
13.
$X$ is normal with $\mu = 40$ and $\sigma = 4$ . Standardize $x = 47$ : what is $z$ ?
Answer: 1.75
$z = \frac{47 - 40}{4} = \frac{7}{4} = 1.75$ . The value 47 lies 1.75 standard deviations above the mean.
14.
$X$ is normal with $\mu = 100$ and $\sigma = 15$ . Find $P(X < 118)$ . Table: $\Phi(1.2) = 0.8849$ , $\Phi(1.8) = 0.9641$ .
Answer: 0.8849
1. What is $x - \mu = 118 - 100$ ? 18
2. What is $z = 18 / 15$ ? 1.2
3. What is $P(Z < 1.2)$ ? 0.8849
15.
$X$ is normal with $\mu = 70$ and $\sigma = 5$ . The table gives $\Phi(1.6) = 0.9452$ . Which line is correct for $P(X > 78)$ ?
1. a) $z = 8$ , area $0$
2. b) $z = -1.6$ , area $0.0548$
3. c) $z = 1.6$ , area $0.0548$
4. d) $z = 1.6$ , area $0.9452$
Answer: $z = 1.6$ , area $0.0548$
$z = \frac{78 - 70}{5} = 1.6$ . The question asks for a right area: $1 - 0.9452 = 0.0548$ . $0.9452$ is the left area, $z = -1.6$ has the wrong sign, and $z = 8$ forgets to divide by $\sigma$ .
16.
$X$ is normal with $\mu = 20$ and $\sigma = 2$ . Find $P(19 < X < 24)$ . Table: $\Phi(0.5) = 0.6915$ , $\Phi(2) = 0.9772$ .
Answer: 0.6687
1. Standardize the lower bound: $z_1 = (19 - 20) / 2$ ? -0.5
2. Standardize the upper bound: $z_2 = (24 - 20) / 2$ ? 2
3. What is $\Phi(-0.5) = 1 - 0.6915$ ? 0.3085
4. What is $0.9772 - 0.3085$ ? 0.6687
17.
The table gives $\Phi(1.96) = 0.975$ . Find $P(-1.96 < Z < 1.96)$ .
Answer: 0.95
$\Phi(-1.96) = 1 - 0.975 = 0.025$ , so $P(-1.96 < Z < 1.96) = 0.975 - 0.025 = 0.95$ . That is why $\pm 1.96$ appears in 95% confidence intervals.
18.
Find the area outside $\pm 1.1$ : $P(Z < -1.1 \text{ or } Z > 1.1)$ . Table: $\Phi(1.1) = 0.8643$ .
Answer: 0.2714
1. What is the right tail $P(Z > 1.1) = 1 - 0.8643$ ? 0.1357
2. By symmetry, what is the left tail $P(Z < -1.1)$ ? 0.1357
3. What is the total outside area? 0.2714