Question

A widget factory wants to do quality control on their gizmos. Gizmos are expected to have a normal distribution with a mean of 250 grams and a standard deviation of 5 grams.

(a) What is the probability of a random gizmo weighing more than 242 grams?

(b) What is the probability that a random gizmo weights between 248 and 252.5 grams?

(c) Find the 84, 97.5, and 99.87-th upper percentiles of the data set. How would this help us find the 16, 2.5, and 0.13-th percentiles?

(d) Find the quartiles of your data set.

Answer 1

Answer:

Based on the given information:

• Gizmos are expected to have a normal distribution
• Mean = 250 grams
• Standard deviation = 5 grams

(a) What is the probability of a random gizmo weighing more than 242 grams?

P(X>242) = P(Z>1.6) = 0.9452

Therefore, probability that a random gizo weighs more than 242 grams is P-value = 0.9452

(b) What is the probability that a random gizmo weights between 248 and 252.5 grams?

Therefore, probability that a random gizo weighs between 248 and 252.5grams is P-value = 0.3469

(c) Find the 84, 97.5, and 99.87-th upper percentiles of the data set. How would this help us find the 16, 2.5, and 0.13-th percentiles?

84th upper percentile corresponds to p =0.84, this is 1 standard deviation away from mean = 250 + 5 = 255

97.5 upper percentile corresponds to p = 0.975, this is 2 standard deviation away from mean = 250 + 10 = 260

99.87 upper percentile corresponds to p = 0.9987, this is 3 standard deviation away from mean = 250 + 15 = 265

16th percentile = 1 standard deviation below from mean = 250 - 5 =245

2.5th percentile = 2 standard deviation below from mean = 250 - 10 = 240

0.13-th percentiles = 3 standard deviation below from mean =250 - 15 = 235

(d) Find the quartiles of your data set.