A particular fruit's weights are normally distributed, with a mean of 593 grams and a standard deviation of 14 grams.
If you pick one fruit at random, what is the probability that it will weigh between 546 grams and 601 grams
To find the probability that a randomly picked fruit will weigh between 546 grams and 601 grams, we need to calculate the area under the normal distribution curve within this range.
To do this, we need to standardize the values of 546 and 601 using the formula:
z = (x - μ) / σ
Where:
x is the value we are standardizing,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
For 546 grams:
z1 = (546 - 593) / 14
z1 ≈ -3.36
For 601 grams:
z2 = (601 - 593) / 14
z2 ≈ 0.57
Now, we need to find the area between these two z-scores on a standard normal distribution table or using a calculator that can compute normal probabilities.
Using a standard normal distribution table, we can find the values corresponding to z1 and z2. The probability, in this case, is the value of z2 minus the value of z1:
P(546 ≤ x ≤ 601) = P(z1 ≤ z ≤ z2)
From the table, the value corresponding to z1 ≈ -3.36 is approximately 0.0004, and the value corresponding to z2 ≈ 0.57 is approximately 0.7123.
Therefore, the probability that a randomly picked fruit will weigh between 546 grams and 601 grams is approximately:
P(546 ≤ x ≤ 601) ≈ 0.7123 - 0.0004
P(546 ≤ x ≤ 601) ≈ 0.7119
So, the probability is approximately 0.7119 or 71.19%.