To solve the problem of finding the probability that a randomly selected value from a normal distribution with a mean of 239.5 and a standard deviation of 27 is greater than 258.4, follow these steps:
1. Identify the parameters of the normal distribution:
- The mean ([tex]\(\mu\)[/tex]) is 239.5.
- The standard deviation ([tex]\(\sigma\)[/tex]) is 27.
2. Determine the value for which you need to find the probability:
- The value [tex]\(x = 258.4\)[/tex].
3. Calculate the z-score for the value 258.4:
The z-score formula for a value [tex]\(x\)[/tex] in a normal distribution is given by:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
Plugging in the numbers:
[tex]\[
z = \frac{258.4 - 239.5}{27} = \frac{18.9}{27} \approx 0.7
\][/tex]
4. Find the cumulative probability for the calculated z-score using the standard normal distribution table or a cumulative distribution function (CDF):
The cumulative probability [tex]\(P(Z \leq 0.7)\)[/tex] gives the probability that a standard normal variable is less than or equal to 0.7.
5. Calculate the cumulative probability for the z-score 0.7.
The cumulative distribution function value for [tex]\(z = 0.7\)[/tex] is approximately 0.7580. This represents the probability [tex]\(P(X \leq 258.4)\)[/tex].
6. Determine the probability that [tex]\(X\)[/tex] is greater than 258.4:
Since the total area under the normal distribution curve is 1, the area to the right of [tex]\(z = 0.7\)[/tex] is:
[tex]\[
P(X > 258.4) = 1 - P(X \leq 258.4) = 1 - 0.7580 = 0.2420
\][/tex]
Therefore, the probability that a randomly selected value from the distribution is greater than 258.4 is:
[tex]\[
P(X > 258.4) = 0.2420
\][/tex]
