Answer :
To determine the probability \(P(x \geq 92)\) for a normal distribution with a mean of 98 and a standard deviation of 6, follow these steps:
1. Identify the given parameters:
- Mean (\(\mu\)) = 98
- Standard deviation (\(\sigma\)) = 6
- Value of \(x\) = 92
2. Compute the z-score:
The z-score helps us determine how many standard deviations away \(x\) is from the mean. The formula for the z-score is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
Plugging in the given values:
[tex]\[ z = \frac{92 - 98}{6} = \frac{-6}{6} = -1 \][/tex]
3. Find the cumulative probability corresponding to the z-score:
Using standard normal distribution tables or a cumulative distribution function (CDF) calculator, we find the cumulative probability for \(z = -1\). This cumulative probability \(P(X < 92)\) is approximately 0.1587.
4. Calculate \(P(x \geq 92)\):
The probability \(P(x \geq 92)\) is the complement of \(P(X < 92)\):
[tex]\[ P(x \geq 92) = 1 - P(X < 92) \][/tex]
Substituting the cumulative probability:
[tex]\[ P(x \geq 92) = 1 - 0.1587 = 0.8413 \][/tex]
Therefore, the probability \(P(x \geq 92)\) is approximately 0.8413.
Among the options provided, the closest value is:
C. 0.84
1. Identify the given parameters:
- Mean (\(\mu\)) = 98
- Standard deviation (\(\sigma\)) = 6
- Value of \(x\) = 92
2. Compute the z-score:
The z-score helps us determine how many standard deviations away \(x\) is from the mean. The formula for the z-score is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
Plugging in the given values:
[tex]\[ z = \frac{92 - 98}{6} = \frac{-6}{6} = -1 \][/tex]
3. Find the cumulative probability corresponding to the z-score:
Using standard normal distribution tables or a cumulative distribution function (CDF) calculator, we find the cumulative probability for \(z = -1\). This cumulative probability \(P(X < 92)\) is approximately 0.1587.
4. Calculate \(P(x \geq 92)\):
The probability \(P(x \geq 92)\) is the complement of \(P(X < 92)\):
[tex]\[ P(x \geq 92) = 1 - P(X < 92) \][/tex]
Substituting the cumulative probability:
[tex]\[ P(x \geq 92) = 1 - 0.1587 = 0.8413 \][/tex]
Therefore, the probability \(P(x \geq 92)\) is approximately 0.8413.
Among the options provided, the closest value is:
C. 0.84
