Question

select the correct answer. the number of tomatoes sold weekly by a grocery store follows a normal distribution with a mean of 455 and a standard deviation of 65. which sentence summarizes this information? a. there is a 68% chance that the grocery store sells fewer than 390 tomatoes in a week. b. there is a 95% chance that the grocery store sells more than 390 tomatoes in a week. c. there is an 84% chance that the grocery store sells more than 390 tomatoes in a week. d. there is a 34% chance that the grocery store sells fewer than 390 tomatoes in a week. reset next

Explanation:

Step1: Calculate the z-score

The z - score formula is $z=\frac{x - \mu}{\sigma}$, where $\mu = 455$ (mean), $\sigma=65$ (standard deviation), and $x = 390$.
Substitute the values: $z=\frac{390 - 455}{65}=\frac{- 65}{65}=- 1$.

Step2: Use the empirical rule for normal distribution

The empirical rule states that:

• Approximately 68% of the data lies within $\mu\pm\sigma$ (i.e., between $455 - 65=390$ and $455 + 65 = 520$).
• Approximately 95% of the data lies within $\mu\pm2\sigma$ (between $455-130 = 325$ and $455 + 130=585$).
• Approximately 99.7% of the data lies within $\mu\pm3\sigma$ (between $455-195 = 260$ and $455 + 195=650$).

For a z - score of $z=-1$, we know that the area to the left of $z=-1$ (probability of selling fewer than 390 tomatoes) is $0.16$ (since the area to the left of $\mu-\sigma$ is $\frac{1 - 0.68}{2}=0.16$), and the area to the right of $z=-1$ (probability of selling more than 390 tomatoes) is $1 - 0.16 = 0.84$ or 84%.

Now let's analyze each option:

• Option A: The probability of selling fewer than 390 is 16%, not 68%. So A is incorrect.
• Option B: The probability of selling more than 390 is 84%, not 95%. So B is incorrect.
• Option C: The probability of selling more than 390 is $1 - 0.16=0.84 = 84\%$. So C is correct.
• Option D: The probability of selling fewer than 390 is 16%, not 34%. So D is incorrect.

Answer:

C. There is an 84% chance that the grocery store sells more than 390 tomatoes in a week.