To calculate the probability that a randomly selected fruit will weigh between 262 grams and 281 grams, we can use the properties of the normal distribution.
Given the following parameters:
- Mean (\(\mu\)): 250 grams
- Standard deviation (\(\sigma\)): 19 grams
To find this probability, we will:
- Convert the weights into z-scores using the formula:
\[ z = \frac{X - \mu}{\sigma} \]
- Use the z-scores to find the corresponding cumulative probabilities from the standard normal distribution table.
Step 1: Calculate z-scores
For \(X = 262\) grams:
\[ z_1 = \frac{262 - 250}{19} = \frac{12}{19} \approx 0.6316 \]
For \(X = 281\) grams:
\[ z_2 = \frac{281 - 250}{19} = \frac{31}{19} \approx 1.6316 \]
Step 2: Find cumulative probabilities
Using a standard normal distribution table (or a calculator), we find:
-
The cumulative probability for \(z_1\) (approximately 0.63):
- \(P(Z < 0.63) \approx 0.7357\)
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The cumulative probability for \(z_2\) (approximately 1.63):
- \(P(Z < 1.63) \approx 0.9484\)
Step 3: Calculate the probability between the two z-scores
To find the probability that the weight of the fruit is between 262 grams and 281 grams, we subtract the cumulative probabilities:
\[ P(262 < X < 281) = P(Z < 1.6316) - P(Z < 0.6316) \approx 0.9484 - 0.7357 \]
\[ P(262 < X < 281) \approx 0.2127 \]
Thus, the probability that a randomly selected fruit will weigh between 262 grams and 281 grams is approximately 0.2127, or 21.27%.
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