Question

bottles of cranberry juice are filled automatically by a machine. the fill volume per bottle is known to follow a normal distribution with mean 600 ml and standard deviation 6 ml. in a random sample of 500 bottles, approximately how many do we expect to contain less than 592 ml? (a) 60 (b) 32 (c) 46 (d) 89 (e) 75

Answer 1

To find how many bottles contain less than 592 ml, calculate the z-score for 592 ml, find the corresponding percentage from a standard normal distribution table, and multiply the probability by the number of bottles (500). The expected number of bottles with less than 592 ml is approximately 46. Option c

Step-by-step explanation:

To determine approximately how many bottles contain less than 592 ml of cranberry juice, we need to calculate the z-score for 592 ml and then find the corresponding percentage from the standard normal distribution table. The formula for z-score is given by:

Z = (X - μ) / σ

Where X is the value we're comparing to, μ is the mean, and σ is the standard deviation. For our case:

Z = (592 - 600) / 6 = -8 / 6 = -1.33

Now, we look up the z-score of -1.33 in the standard normal distribution table, which typically gives us the area to the left of our z-score. The area represents the probability of a bottle having less than 592 ml. The table value for a z-score of -1.33 is approximately 0.0918, or 9.18%.

Finally, to find out the expected number of bottles in the sample of 500 that contain less than 592 ml, we multiply the total number of bottles (500) by the probability:

500 × 0.0918 ≈ 46

Therefore, we expect approximately 46 bottles to contain less than 592 ml. Option c