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Lesson Example Discussion Quiz: Class Homework

Step-4

Title: Normal distribution

Grade: 9-a Lesson: S4-L9

Explanation: Hello Students, time to practice and review the steps for the problem.

Step	Type	Explanation	Answer
1	Problem	X is a normally distributed variable with mean μ = 30 and standard deviation σ = 4. Find a) P(x < 40) b) P(x > 21) c) P(30 < x < 35)
2	Formula:	\$Z = (X-\mu )/ \sigma\$
3	Step	From the given data	\$\mu = 30 , \sigma = 4\$
4	Step	a) P(x < 40) Converting to standard normal curve	\$P(x < 40) = P((X-\mu )/ \sigma) < (40-30 )/ 4)\$ \$P(x < 40) = P(Z < 2.5)\$
5	Step	P(Z < 2.5)	[area to the left of z=2.5]
6	Hint	Take the values from the standard normal distribution table or chart
7	Step	P(Z < 2.5)	0.9938
8	Step	*P(x < 40)	0.9938*
9	Step	b) P(x > 21) Converting to standard normal curve	\$P(x > 21) = P((X-\mu )/ \sigma) > (21-30 )/ 4)\$ \$P(x > 21) = P(Z > -2.25)\$
10	Step	P(Z > -2.25)	[total area]-[area to the left of z=-2.25]
11	Hint	Take the values from the standard normal distribution table or chart
12	Step	P(Z > -2.25)	1-0.0122
13	Step	P(Z > -2.25)	0.9878
14	Step	P(x > 21)	0.9878
15	Step	c) P(30 < x < 35) Converting to standard normal curve	\$P(30 < x < 35) = (30-30 )/ 4 < P((X-\mu )/ \sigma) < (35-30 )/ 4)\$ \$P(30 < x < 35) = P(0 < Z < 1.25)\$
16	Step	P(0 < Z < 1.25)	[area to the left of z=1.25] - [area to the left of 0]
17	Hint	Take the values from the standard normal distribution table or chart
18	Step	P(0 < Z < 1.25)	0.8944 - 0.5
19	Step	P(0 < Z < 1.25)	0.3944
20	Step	P(30 < x < 35)	0.3944
21		Answer	D

Step

Type

Explanation

Answer

Problem

X is a normally distributed variable with mean μ = 30 and standard deviation σ = 4.
Find
a) P(x < 40)
b) P(x > 21)
c) P(30 < x < 35)

Formula:

\$Z = (X-\mu )/ \sigma\$

Step

From the given data

\$\mu = 30 , \sigma = 4\$

Step

a) P(x < 40)
Converting to standard normal curve

\$P(x < 40) = P((X-\mu )/ \sigma) < (40-30 )/ 4)\$

\$P(x < 40) = P(Z < 2.5)\$

Step

P(Z < 2.5)

[area to the left of z=2.5]

Hint

Take the values from the standard normal distribution table or chart

Step

P(Z < 2.5)

0.9938

Step

*P(x < 40)

0.9938*

Step

b) P(x > 21)
Converting to standard normal curve

\$P(x > 21) = P((X-\mu )/ \sigma) > (21-30 )/ 4)\$

\$P(x > 21) = P(Z > -2.25)\$

Step

P(Z > -2.25)

[total area]-[area to the left of z=-2.25]

Hint

Take the values from the standard normal distribution table or chart

Step

P(Z > -2.25)

1-0.0122

Step

P(Z > -2.25)

0.9878

Step

P(x > 21)

0.9878

Step

c) P(30 < x < 35)
Converting to standard normal curve

\$P(30 < x < 35) = (30-30 )/ 4 < P((X-\mu )/ \sigma) < (35-30 )/ 4)\$

\$P(30 < x < 35) = P(0 < Z < 1.25)\$

Step

P(0 < Z < 1.25)

[area to the left of z=1.25] - [area to the left of 0]

Hint

Take the values from the standard normal distribution table or chart

Step

P(0 < Z < 1.25)

0.8944 - 0.5

Step

P(0 < Z < 1.25)

0.3944

Step

P(30 < x < 35)

0.3944

Answer

Tutor: Questions

Seq	Type	Question
1	Problem	What did you learn from this problem?
2	Clue	What did you learn from the clues?
3	Hint	What did you learn from the Hints?
4	Step	What did you learn from the Steps?
5	Step	How can we improve the Steps?

Seq

Type

Question

Audio

Problem

What did you learn from this problem?

Clue

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Hint

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Step

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Step

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