Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever...

Question

Question

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever appropriate. (Do this on paper. Your instructor may ask you to turn in this work.)

(a) P(0 Image for Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wheneve Z 2.74)

(b) P(0 Z 1)

(c) P(-2.40 Z 0)

(d) P(-2.40 Z +2.40)

(e) P(Z 1.63)

(f) P(-1.74 Z)

(g) P(-1.4 Z 2.00)

(h) P(1.63 Z 2.50)

(i) P(1.4 Image for Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wheneve Z)

(j) P( |Z| 2.50)

math Statistics-And-Probability

Add a comment Improve this question Transcribed image text

Answer 1

✔ Recommended Answer

Answer #1

Concepts and reason

A Z-score indicates the number of standard deviations an element is deviated from the mean. Z-scores may be positive or negative. The positive value indicates that the score is above the mean value. The negative value indicates that the score is below the mean value. The normal probability values can be determined with the help of Z-score.

The problem deals with the concepts of finding the normal area probabilities using the known z-scores.

Fundamentals

The probabilities for the standard normal distribution are given in table of areas under the normal distribution or using the Excel function $\left( {{\rm{ = NORM}}{\rm{.S}}{\rm{.DIST(}}z{\rm{,TRUE)}}} \right)$

The table gives the probabilities of the form $P\left( {Z < z} \right).$

(a)

The shaded area in the normal curve with $z = 0$ and $z = 2.74$ is labeled.

0.4969
2.74

The area to the left of $z = 2.74$ is 0.9969.

The area to the left of $z = 0$ is 0.5

The area between $z = 0$ and $z = 2.74$ is,

$\begin{array}{c}\\P\left( {0 < z < 2.74} \right) = 0.9969 - 0.5000\\\\ = 0.4969\\\end{array}$

(b)

The shaded area in the normal curve with $z = 0$ and $z = 1$ is labeled.

10.3413
01

The area to the left of $z = 1$ is 0.8413.

The area to the left of $z = 0$ is 0.5.

The area between $z = 0$ and $z = 1$ is,

$\begin{array}{c}\\P\left( {0 < z < 1} \right) = 0.8413 - 0.5000\\\\ = 0.3413\\\end{array}$

(c)

The shaded area in the normal curve with $z = - 2.40$ and $z = 0$ is labeled.

0.4918
-2.4

The area to the left of $z = - 2.40$ is 0.0082.

The area to the left of $z = 0$ is 0.5000.

The area between $z = - 2.40$ and $z = 0$ is,

$\begin{array}{c}\\P\left( { - 2.40 < z < 0} \right) = P\left( {z < 0} \right) - P\left( {z < - 2.40} \right)\\\\ = 0.5000 - 0.0082\\\\ = 0.4918\\\end{array}$

(d)

The shaded area in the normal curve with $z = - 2.40$ and $z = 2.40$ is labeled.

0.9836
-2.4
2.4

The area to the left of $z = - 2.40$ is 0.0082.

The area to the left of $z = 2.40$ is 0.0.9918.

The area between $z = - 2.40$ and $z = 2.40$ is,

$\begin{array}{c}\\P\left( { - 2.40 < z < 2.40} \right) = P\left( {z < 2.40} \right) - P\left( {z < - 2.40} \right)\\\\ = 0.9918 - 0.0082\\\\ = 0.9836\\\end{array}$

(e)

The shaded area in the normal curve with $z = 1.63$ is labeled.

0.9484
1.63

The area to the left of $z = 1.63$ is 0.9484.

That is,

$P\left( {z \le 1.63} \right) = 0.9484$

(f)

The shaded area in the normal curve with $z = - 1.74$ is labeled.

0.9591
-1.740

The area to the right of $z = - 1.74$ is 0.9591.

That is,

$\begin{array}{c}\\P\left( { - 1.74 \le z} \right) = P\left( {z \ge - 1.74} \right)\\\\ = 1 - P\left( {z < - 1.74} \right)\\\\ = 1 - 0.0409\\\\ = 0.9591\\\end{array}$

(g)

The shaded area in the normal curve with $z = - 1.40$ and $z = 2.00$ is labeled.

10.8964
-1.
4
0

The area to the left of $z = - 1.40$ is 0.0808.

The area to the left of $z = 2$ is 0.9772.

The area between $z = - 1.40$ and $z = 2.00$ is,

$\begin{array}{c}\\P\left( { - 1.40 < z < 2.00} \right) = P\left( {z < 2.00} \right) - P\left( {z < - 1.40} \right)\\\\ = 0.9772 - 0.0808\\\\ = 0.8964\\\end{array}$

(h)

The shaded area in the normal curve with $z = 1.63$ and $z = 2.50$ is labeled.

(0.0453
11.63 2.5

The area to the left of $z = 1.63$ is 0.9484.

The area to the left of $z = 2.50$ is 0.9938.

The area between $z = 1.63$ and $z = 2.50$ is,

$\begin{array}{c}\\P\left( {1.63 < z < 2.50} \right) = P\left( {z < 2.50} \right) - P\left( {z < 1.63} \right)\\\\ = 0.9938 - 0.9484\\\\ = 0.0453\\\end{array}$

(i)

The shaded area in the normal curve with $z = 1.4$ is labeled.

0.0808
01.4

The area to the right of $z = 1.40$ is 0.0808.

That is,

$\begin{array}{c}\\P\left( {1.4 \le z} \right) = P\left( {z \ge 1.4} \right)\\\\ = 1 - P\left( {z < 1.4} \right)\\\\ = 1 - 0.9192\\\\ = 0.0808\\\end{array}$

(j)

The shaded area in the normal curve with $z = - 2.50$ and $z = 2.50$ is labeled.

10.9876
-2.5

The area to the left of $z = - 2.50$ is 0.0062.

The area to the left of $z = 2.50$ is 0.9938.

The area between $z = - 2.50$ and $z = 2.50$ is,

$\begin{array}{c}\\P\left( { - 2.50 < z < 2.50} \right) = P\left( {z < 2.50} \right) - P\left( {z < - 2.50} \right)\\\\ = 0.9938 - 0.0062\\\\ = 0.9876\\\end{array}$

Ans: Part a

Therefore, the area between $z = 0$ and $z = 2.74$ is 0.4969.

Part b

Therefore, the area between $z = 0$ and $z = 1$ is 0.3413.

Part c

Therefore, the area between $z = - 2.40$ and $z = 0$ is 0.4918.

Part d

Therefore, the area between $z = - 2.40$ and $z = 2.40$ is 0.9836.

Part e

Therefore, the area to the left of 1.63 is 0.9484.

Part f

Therefore, the area to the right of $z = - 1.74$ is 0.9591.

Part g

Therefore, the area between $z = - 1.40$ and $z = 2.00$ is 0.8964.

Part h

Therefore, the area between $z = 1.63$ and $z = 2.50$ is 0.0453.

Part i

Therefore, the area to the right of $z = 1.40$ is 0.0808.

Part j

Therefore, the area between $z = - 2.50$ and $z = 2.50$ is 0.9876.

Add a comment

Answer 2

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever...

Homework Answers

Add Answer to:
Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever...

Post as a guest

Earn Coins

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever...

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever...

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever...

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever...

www.webassignnetweb Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures...

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate. a. P(0 ≤ Z ≤ 2.17) h. P(1.37 ≤ Z ≤ 2.50) i. P(1.50 ≤ Z) j. P(|Z| ≤ 2.50)

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever...

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever...

-140 points DevoreStat9 4.E.028 Let Z be a standard normal random variable and calculate the following...

[6] Let z be a standard normal random variable. Compute the following probabilities. P(–1.23 ≤ z...

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever...

Homework Answers

Add Answer to: Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever...

Post as a guest

Earn Coins

Add Answer to:
Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures whenever...