Questions: The weights of steers in a herd are distributed normally. The standard deviation is 200 lbs and the mean steer weight is 1000 lbs. Find the probability that the weight

The weights of steers in a herd are distributed normally. The standard deviation is 200 lbs and the mean steer weight is 1000 lbs. Find the probability that the weight
Transcript text: The weights of steers in a herd are distributed normally. The standard deviation is 200 lbs and the mean steer weight is $\mathbf{1 0 0 0} \mathrm{lbs}$. Find the probability that the weight
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Solution

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Solution Steps

Step 1: Define the Parameters

The weights of steers in a herd are normally distributed with a mean \( \mu = 1000 \) lbs and a standard deviation \( \sigma = 200 \) lbs.

Step 2: Identify the Weight Value

We are interested in finding the probability that the weight of a steer is less than or equal to \( x = 1200 \) lbs.

Step 3: Calculate the Cumulative Distribution Function (CDF)

To find the probability \( P(X \leq 1200) \), we compute the cumulative distribution function (CDF) for the normal distribution at \( x = 1200 \): \[ P(X \leq 1200) = CDF(1200, \mu = 1000, \sigma = 200) \]

Step 4: Evaluate the CDF

The calculated probability is: \[ P(X \leq 1200) \approx 0.8413 \]

Step 5: Interpret the Result

This result indicates that there is approximately an \( 84.13\% \) chance that a randomly selected steer from this herd will weigh less than or equal to \( 1200 \) lbs.

Final Answer

\(\boxed{0.8413}\)

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