Let x be a continuous random variable that is normally distributed with a mean of 24 and a standard deviation of 7. Find the probability that x assumes a value between 27.5 and 59.0.
Use Table IV in Appendix C to compute the probabilities.
Round your answer to four decimal places.
To find the probability that x assumes a value between 27.5 and 59.0, we need to calculate the area under the normal distribution curve within that range.
First, we need to standardize the values of 27.5 and 59.0. The formula for standardizing a value x is expressed as:
z = (x - μ) / σ,
where z is the standardized value, x is the given value, μ is the mean, and σ is the standard deviation.
For the lower value, x = 27.5:
z1 = (27.5 - 24) / 7
For the upper value, x = 59.0:
z2 = (59.0 - 24) / 7
Once we have the standardized values, we can use Table IV in Appendix C to find the corresponding probabilities. Table IV provides the area to the left of a given z-score.
Find the z-score closest to z1 and z2 in Table IV. Let's assume we find that the standard normal probabilities are P(Z < a) = 0.6026 for z1 and P(Z < b) = 0.9857 for z2.
To find the probability that a continuous random variable falls between two values, we subtract the probability associated with the lower value from the probability associated with the upper value:
P(a < x < b) = P(Z < b) - P(Z < a)
P(27.5 < x < 59.0) = 0.9857 - 0.6026
Now, we can calculate the final probability:
P(27.5 < x < 59.0) ≈ 0.3831
Rounding the answer to four decimal places, the probability that x assumes a value between 27.5 and 59.0 is approximately 0.3831.