A particular fruit's weights are normally distributed, with a mean of 353 grams and a standard deviation of 6 grams.
If you pick one fruit at random, what is the probability that it will weigh between 334 grams and 344 grams?
To find the probability that a fruit weighs between 334 grams and 344 grams, we need to calculate the probability within that range.
Step 1: Calculate the z-score for the lower bound of the range.
z = (x - μ) / σ
where x is the lower bound, μ is the mean, and σ is the standard deviation.
For the lower bound of 334 grams:
z = (334 - 353) / 6
Step 2: Calculate the z-score for the upper bound of the range.
For the upper bound of 344 grams:
z = (344 - 353) / 6
Step 3: Look up the cumulative probability from the z-table for both z-scores.
The table gives the cumulative probability up until a particular z-score.
For the lower bound of 334 grams:
P(z < -3.29) ≈ 0.0005 (using a z-table)
For the upper bound of 344 grams:
P(z < -1.5) ≈ 0.0668 (using a z-table)
Step 4: Calculate the probability between the two z-scores.
P(334 < x < 344) ≈ P(z < -1.5) - P(z < -3.29)
≈ 0.0668 - 0.0005
≈ 0.0663 or 6.63%
Therefore, the probability that a randomly picked fruit weighs between 334 grams and 344 grams is approximately 6.63%.
To calculate the probability that a randomly selected fruit weighs between 334 grams and 344 grams, we need to use the concept of the standard normal distribution.
First, let's convert the weights to z-scores using the formula:
z = (x - μ) / σ
where z is the z-score, x is the weight of the fruit, μ is the mean weight, and σ is the standard deviation.
For the lower limit of 334 grams:
z = (334 - 353) / 6 = -3.17
For the upper limit of 344 grams:
z = (344 - 353) / 6 = -1.5
Once we have the z-scores, we can find the corresponding probabilities using a standard normal distribution table or a statistical software.
The probability of a fruit weighing between 334 grams and 344 grams is equal to the area under the normal curve between the z-scores -3.17 and -1.5.
Using a standard normal distribution table, we can find the probabilities corresponding to these z-scores:
P(Z < -1.5) = 0.0668 (from the table)
P(Z < -3.17) ≈ 0 (since it's beyond the lower limit of the table)
To find the probability between the two z-scores, we subtract the probabilities:
P(-1.5 < Z < -3.17) = P(Z < -1.5) - P(Z < -3.17) ≈ 0.0668 - 0 = 0.0668
Therefore, the probability that a randomly picked fruit will weigh between 334 grams and 344 grams is approximately 0.0668 or 6.68%.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between those two Z scores.