Answer :
To solve the problem of finding [tex]\( P(X \leq 82) \)[/tex] where [tex]\( X \)[/tex] follows a normal distribution with a mean ([tex]\( \mu \)[/tex]) of 89 and a standard deviation ([tex]\( \sigma \)[/tex]) of 7, we will follow these steps:
1. Determine the Z-score:
A Z-score tells us how many standard deviations an element is from the mean. The Z-score can be calculated using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
Given:
[tex]\[ X = 82, \quad \mu = 89, \quad \sigma = 7 \][/tex]
Substituting the values:
[tex]\[ Z = \frac{82 - 89}{7} = \frac{-7}{7} = -1 \][/tex]
2. Find the Probability:
We next find the cumulative probability [tex]\( P(Z \leq -1) \)[/tex]. This is typically done using the standard normal distribution tables or a cumulative distribution function (CDF) calculator.
The cumulative probability corresponding to [tex]\( Z = -1 \)[/tex] is approximately 0.1587.
3. Match the Probability to the Multiple-Choice Options:
We now compare this cumulative probability with the given multiple-choice options:
- A. 0.025
- B. 0.975
- C. 0.84
- D. 0.16
We see that the closest value to 0.1587 is 0.16, which corresponds to option D.
Hence, the probability [tex]\( P(X \leq 82) \)[/tex] is approximately 0.16, and the correct answer is:
D. 0.16
1. Determine the Z-score:
A Z-score tells us how many standard deviations an element is from the mean. The Z-score can be calculated using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
Given:
[tex]\[ X = 82, \quad \mu = 89, \quad \sigma = 7 \][/tex]
Substituting the values:
[tex]\[ Z = \frac{82 - 89}{7} = \frac{-7}{7} = -1 \][/tex]
2. Find the Probability:
We next find the cumulative probability [tex]\( P(Z \leq -1) \)[/tex]. This is typically done using the standard normal distribution tables or a cumulative distribution function (CDF) calculator.
The cumulative probability corresponding to [tex]\( Z = -1 \)[/tex] is approximately 0.1587.
3. Match the Probability to the Multiple-Choice Options:
We now compare this cumulative probability with the given multiple-choice options:
- A. 0.025
- B. 0.975
- C. 0.84
- D. 0.16
We see that the closest value to 0.1587 is 0.16, which corresponds to option D.
Hence, the probability [tex]\( P(X \leq 82) \)[/tex] is approximately 0.16, and the correct answer is:
D. 0.16
