To determine the probability [tex]\( P(x \geq 278) \)[/tex] for a normally distributed variable [tex]\( x \)[/tex] with a mean of 266 and a standard deviation of 12, follow these steps:
1. Identify the given parameters:
- Mean ([tex]\(\mu\)[/tex]) = 266
- Standard deviation ([tex]\(\sigma\)[/tex]) = 12
- Value of interest ([tex]\(x\)[/tex]) = 278
2. Calculate the z-score for [tex]\( x = 278 \)[/tex]:
The z-score formula for a given [tex]\( x \)[/tex] is:
[tex]\[
z = \frac{x - \mu}{\sigma}
\][/tex]
Plugging in the values:
[tex]\[
z = \frac{278 - 266}{12} = \frac{12}{12} = 1.0
\][/tex]
3. Find the cumulative probability for the z-score ( [tex]\( P(Z \leq 1.0) \)[/tex] ):
Using standard normal distribution tables or a calculator, the cumulative probability for [tex]\( z = 1.0 \)[/tex] is approximately 0.8413. This represents the probability that [tex]\( x \leq 278 \)[/tex].
4. Determine the probability [tex]\( P(x \geq 278) \)[/tex]:
Since we need the probability that [tex]\( x \geq 278 \)[/tex], we need to find [tex]\( 1 - P(x \leq 278) \)[/tex]:
[tex]\[
P(x \geq 278) = 1 - P(x \leq 278) = 1 - 0.8413 = 0.1587
\][/tex]
Thus, the probability [tex]\( P(x \geq 278) \)[/tex] is approximately 0.1587, which rounds to 0.16.
Therefore, the correct answer is:
[tex]\[ \boxed{0.16} \][/tex]
