Questions: Problem 10: The weights of newborn babies in a certain region follow a normal distribution with a mean of 7.5 pounds and a standard deviation of 1.2 pounds. What is the probability that a randomly selected baby weighs between 6 and 9 pounds?

Problem 10: The weights of newborn babies in a certain region follow a normal distribution with a mean of 7.5 pounds and a standard deviation of 1.2 pounds. What is the probability that a randomly selected baby weighs between 6 and 9 pounds?

Transcript text: Problem 10: The weights of newborn babies in a certain region follow a normal distribution with a mean of 7.5 pounds and a standard deviation of 1.2 pounds. What is the probability that a randomly selected baby weighs between 6 and 9 pounds?

Solution

Solution Steps

Step 1: Define the Normal Distribution Parameters

The weights of newborn babies in the region follow a normal distribution characterized by a mean (\( \mu \)) of 7.5 pounds and a standard deviation (\( \sigma \)) of 1.2 pounds.

Step 2: Calculate the Cumulative Distribution Function (CDF) at 9 Pounds

To find the probability that a randomly selected baby weighs less than or equal to 9 pounds, we compute the CDF at \( x = 9 \): \[ P(X \leq 9) = CDF(9) \approx 0.8944 \]

Step 3: Calculate the Cumulative Distribution Function (CDF) at 6 Pounds

Next, we calculate the CDF at \( x = 6 \): \[ P(X \leq 6) = CDF(6) \approx 0.1056 \]

Step 4: Determine the Probability of Weighing Between 6 and 9 Pounds

The probability that a randomly selected baby weighs between 6 and 9 pounds is given by the difference of the two CDF values: \[ P(6 < X < 9) = P(X \leq 9) - P(X \leq 6) \approx 0.8944 - 0.1056 = 0.7887 \]