Solved on Mar 15, 2024

Find the probability that a randomly selected portable MP3 player from Company XYZ will have a replacement time less than 1.2 years, given the replacement times are normally distributed with $\mu=3.7$ years and $\sigma=1$ year.

STEP 1

1. The replacement times for the portable MP3 players are normally distributed.
2. The mean ( $\mu$ ) of the replacement times is 3.7 years.
3. The standard deviation ( $\sigma$ ) of the replacement times is 1 year.
4. We are looking for the probability that a replacement time is less than 1.2 years, which can be found using the standard normal distribution and $z$ -scores.

STEP 2

1. Calculate the $z$ -score for the replacement time of 1.2 years.
2. Use the standard normal distribution to find the probability corresponding to the calculated $z$ -score.
3. Report the probability accurate to 4 decimal places.

STEP 3

Calculate the $z$ -score for a replacement time of 1.2 years using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 1.2$ years, $\mu = 3.7$ years, and $\sigma = 1$ year.

STEP 4

Substitute the given values into the $z$ -score formula:
$z = \frac{1.2 - 3.7}{1}$

STEP 5

Perform the subtraction in the numerator:
$z = \frac{-2.5}{1}$

STEP 6

Simplify the fraction to find the $z$ -score:
$z = -2.5$

STEP 7

Use the standard normal distribution table, a calculator, or software to find the probability that $Z < -2.5$ , where $Z$ is a standard normal random variable.
This probability is denoted as $P(Z < -2.5)$ .

STEP 8

Look up the probability in the standard normal distribution table or calculate it using technology:
$P(Z < -2.5) \approx 0.0062$

STEP 9

Report the probability accurate to 4 decimal places:
$P(X < 1.2 \text{ years }) = P(Z < -2.5) \approx 0.0062$
The probability that a randomly selected portable MP3 player will have a replacement time less than 1.2 years is approximately 0.0062.

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Find the probability that a randomly selected portable MP3 player from Company XYZ will have a replacement time less than 1.2 years, given the replacement times are normally distributed with $\mu=3.7$ years and $\sigma=1$ year.

STEP 1

STEP 2

1. Calculate the $z$ -score for the replacement time of 1.2 years.
2. Use the standard normal distribution to find the probability corresponding to the calculated $z$ -score.
3. Report the probability accurate to 4 decimal places.

STEP 3

Calculate the $z$ -score for a replacement time of 1.2 years using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 1.2$ years, $\mu = 3.7$ years, and $\sigma = 1$ year.

STEP 4

Substitute the given values into the $z$ -score formula:
$z = \frac{1.2 - 3.7}{1}$

STEP 5

Perform the subtraction in the numerator:
$z = \frac{-2.5}{1}$

STEP 6

Simplify the fraction to find the $z$ -score:
$z = -2.5$

STEP 7

Use the standard normal distribution table, a calculator, or software to find the probability that $Z < -2.5$ , where $Z$ is a standard normal random variable.
This probability is denoted as $P(Z < -2.5)$ .

STEP 8

Look up the probability in the standard normal distribution table or calculate it using technology:
$P(Z < -2.5) \approx 0.0062$

STEP 9

Report the probability accurate to 4 decimal places:
$P(X < 1.2 \text{ years }) = P(Z < -2.5) \approx 0.0062$
The probability that a randomly selected portable MP3 player will have a replacement time less than 1.2 years is approximately 0.0062.

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Find the probability that a randomly selected portable MP3 player from Company XYZ will have a replacement time less than 1.2 years, given the replacement times are normally distributed with μ=3.7\mu=3.7μ=3.7 years and σ=1\sigma=1σ=1 year.

STEP 1

STEP 2

1. Calculate the zzz-score for the replacement time of 1.2 years.2. Use the standard normal distribution to find the probability corresponding to the calculated zzz-score.3. Report the probability accurate to 4 decimal places.

STEP 3

Calculate the zzz-score for a replacement time of 1.2 years using the formula:z=X−μσ z = \frac{X - \mu}{\sigma} z=σX−μ​where X=1.2X = 1.2X=1.2 years, μ=3.7\mu = 3.7μ=3.7 years, and σ=1\sigma = 1σ=1 year.

STEP 4

Substitute the given values into the zzz-score formula:z=1.2−3.71 z = \frac{1.2 - 3.7}{1} z=11.2−3.7​

STEP 5

Perform the subtraction in the numerator:z=−2.51 z = \frac{-2.5}{1} z=1−2.5​

STEP 6

Simplify the fraction to find the zzz-score:z=−2.5 z = -2.5 z=−2.5

STEP 7

Use the standard normal distribution table, a calculator, or software to find the probability that Z<−2.5Z < -2.5Z<−2.5, where ZZZ is a standard normal random variable.This probability is denoted as P(Z<−2.5)P(Z < -2.5)P(Z<−2.5).

STEP 8

Look up the probability in the standard normal distribution table or calculate it using technology:P(Z<−2.5)≈0.0062 P(Z < -2.5) \approx 0.0062 P(Z<−2.5)≈0.0062

STEP 9

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Find the probability that a randomly selected portable MP3 player from Company XYZ will have a replacement time less than 1.2 years, given the replacement times are normally distributed with $\mu=3.7$ years and $\sigma=1$ year.

1. Calculate the $z$ -score for the replacement time of 1.2 years.
2. Use the standard normal distribution to find the probability corresponding to the calculated $z$ -score.
3. Report the probability accurate to 4 decimal places.

Calculate the $z$ -score for a replacement time of 1.2 years using the formula:
$z = \frac{X - \mu}{\sigma}$
where $X = 1.2$ years, $\mu = 3.7$ years, and $\sigma = 1$ year.

Substitute the given values into the $z$ -score formula:
$z = \frac{1.2 - 3.7}{1}$

Perform the subtraction in the numerator:
$z = \frac{-2.5}{1}$

Simplify the fraction to find the $z$ -score:
$z = -2.5$

Use the standard normal distribution table, a calculator, or software to find the probability that $Z < -2.5$ , where $Z$ is a standard normal random variable.
This probability is denoted as $P(Z < -2.5)$ .

Look up the probability in the standard normal distribution table or calculate it using technology:
$P(Z < -2.5) \approx 0.0062$