Final Answer
The probability that the mean weight of 12 fruits picked at random will be between 692 grams and 701 grams is approximately 0.0548.
Explanation
To find the probability, first, we note that the distribution of the sample means follows a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size (in this case, √12 since there are 12 fruits).
Given the mean (μ) = 704 grams, standard deviation (σ) = 12 grams, and a sample size of 12 fruits, we calculate the standard deviation of the sample mean (σ/√n) as 12/√12 ≈ 3.464.
Next, we standardize the values 692 and 701 using the z-score formula to transform them into z-scores. For 692 grams: z = (692 - 704) / (12 / √12) ≈ -3.464 and for 701 grams: z = (701 - 704) / (12 / √12) ≈ -0.866.
Using a standard normal distribution table or a calculator to find the area under the standard normal curve between these z-scores, we find the probability. To find the probability between -3.464 and -0.866, we subtract the cumulative probability at -3.464 from the cumulative probability at -0.866.
This gives us the probability that the mean weight of 12 randomly picked fruits will fall between 692 and 701 grams, which is approximately 0.0548 when rounded to four decimal places.