Math / Data & Statistics

QuestionThe life of a Radio Shack record player is normally distributed with a mean of 3.3 years and a standard deviation of 0.8 years. Radio Shack guarantees its record players for 2 years.
Find the probability that a record player will break down during the guarantee period. 0.03417 0.05208 0.06302 0.04218 0.03796 0.05729 0.04687

Studdy Solution

STEP 1

1. The life of the record player follows a normal distribution with a mean ( $\mu$ ) of 3.3 years and a standard deviation ( $\sigma$ ) of 0.8 years.
2. We need to find the probability that the record player breaks down within 2 years.

STEP 2

1. Standardize the value using the Z-score formula.
2. Use the standard normal distribution to find the probability.

STEP 3

Calculate the Z-score for 2 years using the formula:
$Z = \frac{X - \mu}{\sigma}$
where $X = 2$ , $\mu = 3.3$ , and $\sigma = 0.8$ .
$Z = \frac{2 - 3.3}{0.8} = \frac{-1.3}{0.8} = -1.625$

STEP 4

Use the Z-score to find the probability from the standard normal distribution table. The Z-score of -1.625 corresponds to the probability that the record player will break down within 2 years.
Look up $Z = -1.625$ in the standard normal distribution table, or use a calculator to find the probability.
$P(Z < -1.625) \approx 0.05208$
The probability that a record player will break down during the guarantee period is:
$\boxed{0.05208}$

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