Question

Assume that the weight of ripe watermelons grown at a particular farm are normally distributed with a mean of 30 pounds and a standard deviation of 2.6 pounds. determine the percent of watermelons that weigh between 26.87 pounds and 31.95 pounds

Answer 1

To find the percent of watermelons weighing between 26.87 and 31.95 pounds, we convert the weights to z-scores and use the standard normal distribution to find that approximately 65.93% of watermelons fall in that range.

Step-by-step explanation:

To calculate the percent of watermelons that weigh between 26.87 pounds and 31.95 pounds, given that the weights are normally distributed with a mean of 30 pounds and a standard deviation of 2.6 pounds, we need to use the normal distribution properties and z-scores.

First, we'll convert the weights into z-scores using the formula:

Z = (X - mean) / standard deviation

For 26.87 pounds:

Z1 = (26.87 - 30) / 2.6 = -1.205

For 31.95 pounds:

Z2 = (31.95 - 30) / 2.6 = 0.750

We then look up these z-scores in a standard normal distribution table or use a calculator to find the probabilities.

P(Z1 < Z < Z2) = P(Z < 0.750) - P(Z < -1.205)

Assume P(Z < 0.750) = 0.7734 and P(Z < -1.205) = 0.1141, then:

P(Z1 < Z < Z2) = 0.7734 - 0.1141 = 0.6593 or 65.93%

Therefore, approximately 65.93% of watermelons weigh between 26.87 pounds and 31.95 pounds.