Question

A particular fruit's weights are normally distributed, with a mean of 430 grams and a standard deviation of 8 If you plick one fruit at random, what is the probability that it will weigh between 415 grams and 417grams grams.

Answer 1

The probability that a randomly picked fruit weighs between 415 grams and 417 grams is approximately 0.0212.

Step-by-step explanation:

In a normal distribution, the probability of an event occurring within a specific range can be calculated using the z-score formula: Z =
`\frac{{X - \mu}}{{\sigma}} \)`, where X is the value,
`\( \mu \)` is the mean, and
`\( \sigma \)` is the standard deviation.

For this problem, we want to find the probability that the fruit's weight falls between 415 grams and 417 grams. First, we calculate the z-scores for both values:

For 415 grams:

`\[ Z_1 = \frac{{415 - 430}}{{8}}` = -1.875

For 417 grams:

`\[ Z_2 = \frac{{417 - 430}}{{8}}` = -1.625

Next, we look up these z-scores in the standard normal distribution table to find the corresponding probabilities. For Z = -1.875 , the probability is approximately 0.0301, and for Z = -1.625 , the probability is approximately 0.0537.

Now, we subtract the smaller probability from the larger one to find the probability of the fruit weighing between 415 grams and 417 grams:

P(415 < X < 417) = P(Z < -1.625) - P(Z < -1.875)

= 0.0537 - 0.0301

≈ 0.0212

Therefore, the probability that a randomly picked fruit weighs between 415 grams and 417 grams is approximately 0.0212.