Math

Question 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=3.7 years and σ=1\sigma=1 year.

Studdy Solution

STEP 1

1. The replacement times for the portable MP3 players are normally distributed.
2. The mean (μ\mu) of the replacement times is 3.7 years.
3. The standard deviation (σ\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 zz-scores.

STEP 2

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

STEP 3

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

STEP 4

Substitute the given values into the zz-score formula:
z=1.23.71 z = \frac{1.2 - 3.7}{1}

STEP 5

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

STEP 6

Simplify the fraction to find the zz-score:
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 < -2.5, where ZZ is a standard normal random variable.
This probability is denoted as 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

STEP 9

Report the probability accurate to 4 decimal places:
P(X<1.2 years )=P(Z<2.5)0.0062 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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