Questions: The weights of steers in a herd are distributed normally. The variance is 40,000 and the mean steer weight is 1000 lbs. Find the probability that the weight of a randomly selected steer is between 819 and 1320 lbs. Round your answer to four decimal places.

The weights of steers in a herd are distributed normally. The variance is 40,000 and the mean steer weight is 1000 lbs. Find the probability that the weight of a randomly selected steer is between 819 and 1320 lbs. Round your answer to four decimal places.

Transcript text: The weights of steers in a herd are distributed normally. The variance is 40,000 and the mean steer weight is 1000 lbs. Find the probability that the weight of a randomly selected steer is between 819 and 1320 lbs. Round your answer to four decimal places.

Solution

Solution Steps

Step 1: Given Information

The weights of steers in a herd are normally distributed with a mean (\( \mu \)) of 1000 lbs and a variance (\( \sigma^2 \)) of 40000. The standard deviation (\( \sigma \)) can be calculated as: \[ \sigma = \sqrt{40000} = 200 \]

Step 2: Calculate CDF for Lower Bound

To find the probability that a randomly selected steer weighs less than 819 lbs, we calculate the cumulative distribution function (CDF) at \( x = 819 \): \[ P(X \leq 819) \approx 0.1827 \]

Step 3: Calculate CDF for Upper Bound

Next, we calculate the CDF at \( x = 1320 \): \[ P(X \leq 1320) \approx 0.9452 \]

Step 4: Calculate Probability Between Bounds

The probability that the weight of a randomly selected steer is between 819 lbs and 1320 lbs is given by: \[ P(819 < X < 1320) = P(X \leq 1320) - P(X \leq 819) \approx 0.9452 - 0.1827 = 0.7625 \]