To determine which sentence accurately summarizes the probability that a store sells more than 390 CDs in a week given a normal distribution with a mean of 455 and a standard deviation of 65, we'll follow these steps:
1. Define the Parameters:
- Mean ([tex]\(\mu\)[/tex]) = 455 CDs
- Standard deviation ([tex]\(\sigma\)[/tex]) = 65 CDs
- Value of interest (X) = 390 CDs
2. Calculate the Z-score:
The Z-score formula is given by:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\(X\)[/tex] is the value of interest (390 CDs)
- [tex]\(\mu\)[/tex] is the mean (455 CDs)
- [tex]\(\sigma\)[/tex] is the standard deviation (65 CDs)
So,
[tex]\[
Z = \frac{390 - 455}{65} = \frac{-65}{65} = -1
\][/tex]
3. Find the Probability:
To find the probability associated with a Z-score of -1, we refer to the cumulative distribution function (CDF) of a standard normal distribution. The CDF gives the probability that a standard normal random variable is less than or equal to a given value.
For [tex]\(Z = -1\)[/tex], the CDF value is approximately 0.1587. This means there is a 15.87% chance that the shop sells fewer than 390 CDs in a week.
4. Complementary Probability:
To find the probability that the shop sells more than 390 CDs, we need to consider the complement of the CDF at [tex]\(Z = -1\)[/tex]. The complementary probability can be calculated as:
[tex]\[
P(X > 390) = 1 - P(X \leq 390)
\][/tex]
So,
[tex]\[
P(X > 390) = 1 - 0.1587 = 0.8413
\][/tex]
This means there is approximately an 84.13% chance that the shop sells more than 390 CDs in a week.
Given this information, the correct answer is:
- A. There is an 84% chance that the shop sells more than 390 CDs in a week.
