Math Snap

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 $\sigma = 0.8$.
$$Z = \frac{2 - 3.3}{0.8} = \frac{-1.3}{0.8} = -1.625$$

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:
$$\boxed{0.05208}$$