Standard Normal Distribution

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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) $\mu = 0$ , $\sigma = 0$
2.
The z-table gives $\Phi(0.5) = 0.6915$ . Find $P(Z > 0.5)$ .
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 $0.09$ .
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 $z = 1$
5.
Find $P(Z < 0)$ for the standard normal variable $Z$ . Give a decimal.
6.
Find $P(Z > 2)$ . The z-table gives $\Phi(2) = 0.9772$ .
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) $0.8849$
8.
Find $P(Z < -0.25)$ using symmetry. The table gives $\Phi(0.25) = 0.5987$ .
9.
The table gives $\Phi(1.75) = 0.9599$ . Find $P(Z < -1.75)$ .
10.
Find $P(0.5 < Z < 1.5)$ . Table: $\Phi(0.5) = 0.6915$ , $\Phi(1.5) = 0.9332$ .
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) $0.6826$
12.
Find $P(-1 < Z < 1)$ with the table value $\Phi(1) = 0.8413$ , then compare with the 68-95-99.7 rule.
13.
$X$ is normal with $\mu = 40$ and $\sigma = 4$ . Standardize $x = 47$ : what is $z$ ?
14.
$X$ is normal with $\mu = 100$ and $\sigma = 15$ . Find $P(X < 118)$ . Table: $\Phi(1.2) = 0.8849$ , $\Phi(1.8) = 0.9641$ .
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 $0.9452$
16.
$X$ is normal with $\mu = 20$ and $\sigma = 2$ . Find $P(19 < X < 24)$ . Table: $\Phi(0.5) = 0.6915$ , $\Phi(2) = 0.9772$ .
17.
The table gives $\Phi(1.96) = 0.975$ . Find $P(-1.96 < Z < 1.96)$ .
18.
Find the area outside $\pm 1.1$ : $P(Z < -1.1 \text{ or } Z > 1.1)$ . Table: $\Phi(1.1) = 0.8643$ .