To determine the probability that a normally distributed random variable [tex]\(x\)[/tex] with a mean [tex]\(\mu = 89\)[/tex] and a standard deviation [tex]\(\sigma = 5\)[/tex] takes on a value less than 87, follow these steps:
1. Standardize the Variable:
Convert the value 87 to its corresponding z-score. The z-score measures how many standard deviations a particular value is from the mean.
The formula for the z-score is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
Plug in the given values:
[tex]\[
z = \frac{87 - 89}{5} = \frac{-2}{5} = -0.4
\][/tex]
2. Find the Cumulative Probability:
The next step is to find the cumulative probability corresponding to the z-score. This cumulative probability represents the area under the standard normal curve to the left of [tex]\(z = -0.4\)[/tex]. This can be found using standard normal distribution tables, calculators, or software tools.
3. Determine the Probability:
Looking up [tex]\(z = -0.4\)[/tex] in the cumulative distribution function (CDF) table for the standard normal distribution, or utilizing a proper calculator or software, we obtain the cumulative probability.
The cumulative probability corresponding to a z-score of -0.4 is approximately 0.3446.
Thus, the probability that [tex]\(x<87\)[/tex] is:
[tex]\[
P(x < 87) \approx 0.3446
\][/tex]
When rounding to four decimal places, the probability [tex]\(P(x < 87)\)[/tex] is [tex]\(0.3446\)[/tex].
