Question
The mean weight of a herd of white-tailed deer is 140.3 pounds, with a standard deviation of 7.2 pounds.
What is the probability that a randomly selected deer weighs more than 149 pounds?
What is the probability that a randomly selected deer weighs more than 149 pounds?
Answers
GPT-4o mini
To find the probability that a randomly selected deer weighs more than 149 pounds, we can use the properties of the normal distribution. Given that the mean weight \(\mu\) is 140.3 pounds and the standard deviation \(\sigma\) is 7.2 pounds, we can standardize the value of 149 pounds using the z-score formula:
\[
z = \frac{X - \mu}{\sigma}
\]
Where:
- \(X\) is the value we are interested in (149 pounds),
- \(\mu\) is the mean (140.3 pounds),
- \(\sigma\) is the standard deviation (7.2 pounds).
Let's calculate the z-score for 149 pounds:
\[
z = \frac{149 - 140.3}{7.2} = \frac{8.7}{7.2} \approx 1.2083
\]
Next, we will use the z-score to find the probability. We want to find \(P(X > 149)\), which is equivalent to finding \(P(Z > 1.2083)\).
Using standard normal distribution tables or calculators, we can find the cumulative probability associated with \(z = 1.2083\).
1. Look up the z-score of 1.2083 in a z-table or use a calculator. The cumulative probability \(P(Z < 1.2083)\) is approximately 0.8869 (This value may vary slightly based on different statistical tables or software).
2. To find the probability of a deer weighing more than 149 pounds, we calculate:
\[
P(X > 149) = 1 - P(Z < 1.2083) = 1 - 0.8869 \approx 0.1131
\]
Thus, the probability that a randomly selected deer weighs more than 149 pounds is approximately \(0.1131\), or 11.31%.
\[
z = \frac{X - \mu}{\sigma}
\]
Where:
- \(X\) is the value we are interested in (149 pounds),
- \(\mu\) is the mean (140.3 pounds),
- \(\sigma\) is the standard deviation (7.2 pounds).
Let's calculate the z-score for 149 pounds:
\[
z = \frac{149 - 140.3}{7.2} = \frac{8.7}{7.2} \approx 1.2083
\]
Next, we will use the z-score to find the probability. We want to find \(P(X > 149)\), which is equivalent to finding \(P(Z > 1.2083)\).
Using standard normal distribution tables or calculators, we can find the cumulative probability associated with \(z = 1.2083\).
1. Look up the z-score of 1.2083 in a z-table or use a calculator. The cumulative probability \(P(Z < 1.2083)\) is approximately 0.8869 (This value may vary slightly based on different statistical tables or software).
2. To find the probability of a deer weighing more than 149 pounds, we calculate:
\[
P(X > 149) = 1 - P(Z < 1.2083) = 1 - 0.8869 \approx 0.1131
\]
Thus, the probability that a randomly selected deer weighs more than 149 pounds is approximately \(0.1131\), or 11.31%.