Question

a juice machine is set to dispense 16 ounces of juice. the amount of juice dispensed is normally distributed, with a mean of 16.15 ounces and a standard deviation of 0.25 ounces. in which range will the amount of juice dispensed be found 99.7% of the time?

a. 15.90 ounces to 16.40 ounces
b. 15.65 ounces to 16.65 ounces
c. 15.40 ounces to 16.90 ounces
d. 15.15 ounces to 17.15 ounces

Explanation:

Step1: Recall the empirical rule for normal distribution

For a normal distribution, approximately 99.7% of the data lies within \( \mu - 3\sigma \) and \( \mu + 3\sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.
Here, \( \mu = 16.15 \) ounces and \( \sigma = 0.25 \) ounces.

Step2: Calculate the lower bound

Lower bound \( = \mu - 3\sigma = 16.15 - 3\times0.25 \)
\( = 16.15 - 0.75 = 15.40 \) ounces? Wait, no, wait, 30.25 is 0.75? Wait, no, 0.253 is 0.75? Wait, no, 16.15 - 30.25: 30.25 is 0.75, 16.15 - 0.75 = 15.40? Wait, but let's check again. Wait, maybe I made a mistake. Wait, 16.15 - 30.25: 0.253 is 0.75, 16.15 - 0.75 = 15.40? But let's check the options. Wait, option B is 15.65 to 16.65, which is \( \mu - 2\sigma \) and \( \mu + 2\sigma \) (since 16.15 - 20.25 = 16.15 - 0.5 = 15.65, 16.15 + 20.25 = 16.15 + 0.5 = 16.65), which is 95% of the data. Wait, no, the question is 99.7%, so it's 3 standard deviations. Wait, 16.15 - 30.25 = 16.15 - 0.75 = 15.40, 16.15 + 30.25 = 16.15 + 0.75 = 16.90. But wait, the options: option C is 15.40 to 16.90. But wait, let's check the options again. Wait, maybe I miscalculated. Wait, 0.25*3 is 0.75. So 16.15 - 0.75 = 15.40, 16.15 + 0.75 = 16.90. So the range is 15.40 to 16.90, which is option C? Wait, but let's check the options again. Wait, the options are:

A. 15.90 to 16.40 (which is \( \mu - \sigma \) to \( \mu + \sigma \), since 16.15 - 0.25 = 15.90, 16.15 + 0.25 = 16.40, which is 68% of data)

B. 15.65 to 16.65 (which is \( \mu - 2\sigma \) to \( \mu + 2\sigma \), 16.15 - 20.25 = 16.15 - 0.5 = 15.65, 16.15 + 20.25 = 16.15 + 0.5 = 16.65, which is 95% of data)

C. 15.40 to 16.90 (which is \( \mu - 3\sigma \) to \( \mu + 3\sigma \), 16.15 - 30.25 = 16.15 - 0.75 = 15.40, 16.15 + 30.25 = 16.15 + 0.75 = 16.90, which is 99.7% of data)

D. 15.15 to 17.15 (which is \( \mu - 4\sigma \) to \( \mu + 4\sigma \), 16.15 - 40.25 = 16.15 - 1 = 15.15, 16.15 + 40.25 = 16.15 + 1 = 17.15, but 99.7% is within 3σ, not 4σ)

So the correct range is 15.40 ounces to 16.90 ounces, which is option C. Wait, but wait, let's check the calculation again. 3*0.25 is 0.75. 16.15 - 0.75 = 15.40, 16.15 + 0.75 = 16.90. So that's correct.

Answer:

C. 15.40 ounces to 16.90 ounces