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Lesson Example Discussion Quiz: Class Homework

Step-5

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	The length of similar components produced by a company are approximated by a normal distribution model with a mean of 5 cm and a standard deviation of 0.02 cm. If a component is chosen at random a) what is the probability that the length of this component is between 4.98 and 5.02 cm? b) what is the probability that the length of this component is between 4.96 and 5.04 cm?
2	Formula:	\$Z = (X-\mu )/ \sigma\$
3	Clue	a) Probability that the length of this component is between 4.98 and 5.02 cm	P(4.98<X<5.02)
4	Step	a) From the given data	\$\mu = 5 , \sigma = 0.02 , X_1 = 4.98, X_2 = 5.02\$
5	Step	Converting to standard normal curve	\$P(4.98<X<5.02) = (4.98-5)/ 0.02 < P((X-\mu )/ \sigma) < (5.02-5)/ 0.02)\$ \$P(4.98<X<5.02) = P(-1<Z<1)\$
6	Step	P(-1<Z<1)	P(Z<1) - P(Z←1)
7	Hint	Take the values from the standard normal distribution table or chart
8	Step	P(-1<Z<1)	0.8413 - 0.1587
9	Step	P(-1<Z<1)	0.6826
10	Step	a) Probability that the length of this component is between 4.98 and 5.02 cm	0.6826
11	Clue	b) Probability that the length of this component is between 4.96 and 5.04 cm	P(4.96<X<5.04)
12	Step	b) From the given data	\$\mu = 5 , \sigma = 0.02 , X_1 = 4.96, X_2 = 5.04\$
13	Step	Converting to standard normal curve	\$P(4.96<X<5.02) = (4.96-5)/ 0.02 < P((X-\mu )/ \sigma) < (5.04-5)/ 0.02)\$ \$P(4.96<X<5.04) = P(-2<Z<2)\$
14	Step	P(-2<Z<2)	P(Z<2) - P(Z←2)
15	Hint	Take the values from the standard normal distribution table or chart
16	Step	P(-2<Z<2)	0.9772 - 0.0228
17	Step	P(-2<Z<2)	0.9544
18	Step	b) Probability that the length of this component is between 4.96 and 5.04 cm	0.9544
19		Answer	B

Step

Type

Explanation

Answer

Problem

The length of similar components produced by a company are approximated by a normal distribution model with a mean of 5 cm and a standard deviation of 0.02 cm. If a component is chosen at random
a) what is the probability that the length of this component is between 4.98 and 5.02 cm?
b) what is the probability that the length of this component is between 4.96 and 5.04 cm?

Formula:

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

Clue

a) Probability that the length of this component is between 4.98 and 5.02 cm

P(4.98<X<5.02)

Step

a) From the given data

\$\mu = 5 , \sigma = 0.02 , X_1 = 4.98, X_2 = 5.02\$

Step

Converting to standard normal curve

\$P(4.98<X<5.02) = (4.98-5)/ 0.02 < P((X-\mu )/ \sigma) < (5.02-5)/ 0.02)\$

\$P(4.98<X<5.02) = P(-1<Z<1)\$

Step

P(-1<Z<1)

P(Z<1) - P(Z←1)

Hint

Take the values from the standard normal distribution table or chart

Step

P(-1<Z<1)

0.8413 - 0.1587

Step

P(-1<Z<1)

0.6826

Step

a) Probability that the length of this component is between 4.98 and 5.02 cm

0.6826

Clue

b) Probability that the length of this component is between 4.96 and 5.04 cm

P(4.96<X<5.04)

Step

b) From the given data

\$\mu = 5 , \sigma = 0.02 , X_1 = 4.96, X_2 = 5.04\$

Step

Converting to standard normal curve

\$P(4.96<X<5.02) = (4.96-5)/ 0.02 < P((X-\mu )/ \sigma) < (5.04-5)/ 0.02)\$

\$P(4.96<X<5.04) = P(-2<Z<2)\$

Step

P(-2<Z<2)

P(Z<2) - P(Z←2)

Hint

Take the values from the standard normal distribution table or chart

Step

P(-2<Z<2)

0.9772 - 0.0228

Step

P(-2<Z<2)

0.9544

Step

b) Probability that the length of this component is between 4.96 and 5.04 cm

0.9544

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

What did you learn from the clues?

Hint

What did you learn from the Hints?

Step

What did you learn from the Steps?

Step

How can we improve the Steps?