The probability that the average weight of 13 randomly selected babies is between 3000 grams and 3600 grams, rounded to 4 decimal places, is 0.4682.
To solve this problem:
Step 1: Determine the parameters of the normal distribution
The mean of the normal distribution is given as μ = 3320 grams.
The standard deviation of the normal distribution is given as σ = 564 grams.
Step 2: Determine the z-scores of the two weight limits
The z-score of the lower weight limit (3000 grams) is calculated as:
z = (3000 - μ) / σ = (3000 - 3320) / 564 ≈ -0.57
The z-score of the upper weight limit (3600 grams) is calculated as:
z = (3600 - μ) / σ = (3600 - 3320) / 564 ≈ 0.50
Step 3: Calculate the probability using a z-table or calculator
The probability that the average weight of 13 randomly selected babies is between 3000 grams and 3600 grams can be calculated using a z-table or a calculator with a normal distribution function (cdf).
Using a z-table, find the areas between z = -0.57 and z = 0.50. The area between z = -0.57 and z = 0 is 0.2233, and the area between z = 0 and z = 0.50 is 0.6915. Subtracting the two areas, we get:
P(-0.57 < z < 0.50) = 0.6915 - 0.2233 ≈ 0.4682
Step 4: Round the answer to 4 decimal places
Therefore the probability that the average weight of 13 randomly selected babies is between 3000 grams and 3600 grams, rounded to 4 decimal places, is 0.4682.