A distribution of values is normal with a mean of 134.7 and a standard deviation of 61.3 .
Find the probability that a randomly selected value is between -36.9 and 226.7 . Round your answer to 4
decimal places.
probability
Question Help: Video Message instructor

Ask by Long Patel. in the United States

Mar 27,2025

Question

A distribution of values is normal with a mean of 134.7 and a standard deviation of 61.3 .
Find the probability that a randomly selected value is between -36.9 and 226.7 . Round your answer to 4
decimal places.
probability
Question Help: Video Message instructor

Ask by Long Patel. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

1. The variable $$ X $$ follows a normal distribution with mean $$\mu = 134.7$$ and standard deviation $$\sigma = 61.3$$.

2. We need to calculate the probability:
   $$
   P(-36.9 < X < 226.7)
   $$

3. Convert the boundaries to $$ z $$-scores using:
   $$
   z = \frac{X - \mu}{\sigma}
   $$

4. For the lower boundary $$ X = -36.9 $$:
   $$
   z = \frac{-36.9 - 134.7}{61.3} = \frac{-171.6}{61.3} \approx -2.8
   $$

5. For the upper boundary $$ X = 226.7 $$:
   $$
   z = \frac{226.7 - 134.7}{61.3} = \frac{92}{61.3} \approx 1.5
   $$

6. Thus, the required probability is:
   $$
   P(-36.9 < X < 226.7) = P(-2.8 < Z < 1.5)
   $$

7. Using the standard normal cumulative distribution function (CDF), we find:
   $$
   \Phi(1.5) \approx 0.9332 \quad \text{and} \quad \Phi(-2.8) \approx 0.0026
   $$

8. Therefore, the probability is:
   $$
   P(-2.8 < Z < 1.5) = \Phi(1.5) - \Phi(-2.8) \approx 0.9332 - 0.0026 = 0.9306
   $$

9. Rounding to four decimal places, the probability is:
   $$
   \boxed{0.9306}
   $$
The probability that a randomly selected value is between -36.9 and 226.7 is approximately 0.9306.

A distribution of values is normal with a mean of 134.7 and a standard deviation of 61.3 .
Find the probability that a randomly selected value is between -36.9 and 226.7 . Round your answer to 4
decimal places.
probability
Question Help: Video Message instructor

Upstudy AI Solution

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A distribution of values is normal with a mean of 134.7 and a standard deviation of 61.3 . Find the probability that a randomly selected value is between -36.9 and 226.7 . Round your answer to 4 decimal places. probability Question Help: Video Message instructor

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Latest Statistics Questions

A distribution of values is normal with a mean of 134.7 and a standard deviation of 61.3 .
Find the probability that a randomly selected value is between -36.9 and 226.7 . Round your answer to 4
decimal places.
probability
Question Help: Video Message instructor