Question

From a sample of 150 marbles of a production line having mean diameter of 10 mm, and a standard deviation of 3.4mm, how many marbles would you expect to find that their diameters are in the range from 10 to 15 mm (assuming normal distribution)?

Answer 1

To find the number of marbles with diameters between 10 and 15 mm, convert the range to z-scores, find the corresponding probabilities, and multiply the probability by the sample size. Approximately 64 marbles would be expected in that range.

Step-by-step explanation:

To determine the expected number of marbles with diameters in the range from 10 to 15 mm from a normal distribution with a mean diameter of 10 mm and a standard deviation of 3.4 mm, we can calculate the probability of a marble falling within this range and then multiply that probability by the total number of marbles in the sample.

Steps for Calculation:

1. Convert the range to z-scores. The z-score for 10 mm is (10-10)/3.4 = 0 since it is the mean. The z-score for 15 mm is (15-10)/3.4 ≈ 1.47.
2. Find the probability corresponding to these z-scores using a standard normal distribution table or a calculator with normal distribution functions. Probability of z < 1.47 is approximately 0.9292.
3. Subtract the probability of z < 0 (which is 0.5, as 0 is the mean) from the probability of z < 1.47 to find the probability of a marble being between the diameters of 10 mm and 15 mm. The result is 0.9292 - 0.5 = 0.4292.
4. Multiply this probability by the total number of marbles (150) to find the expected number of marbles within this diameter range. So, 150 * 0.4292 ≈ 64.38.

Therefore, we would expect to find approximately 64 marbles with diameters ranging from 10 to 15 mm in the given sample.