Question
\( X \) is a normally distributed random variable with mean 65 and standard deviation 11 . What is the probability that \( X \) is less than 80 ? Write your answer as a decimal rounded to the nearest thousandth.
Ask by Pierce Lewis. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The probability that \( X \) is less than 80 is approximately 0.913.
Solution
To find the probability that a normally distributed random variable \( X \) is less than 80, we can use the z-score formula and a standard normal distribution table.
Given:
- Mean (\( \mu \)) = 65
- Standard Deviation (\( \sigma \)) = 11
- Value of interest (\( X \)) = 80
First, we need to calculate the z-score for \( X = 80 \) using the formula:
\[ z = \frac{X - \mu}{\sigma} \]
Substitute the given values into the formula:
\[ z = \frac{80 - 65}{11} \]
\[ z = \frac{15}{11} \]
\[ z \approx 1.3636 \]
Next, we use the z-score to find the probability that \( X \) is less than 80. We look up the z-score in a standard normal distribution table to find the corresponding probability.
The probability that \( X \) is less than 80 is approximately 0.913.
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Simplify this solution Bonus Knowledge
To find the probability that \( X \) is less than 80, we need to standardize the variable using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] Here, \( \mu = 65 \) and \( \sigma = 11 \). Calculating the Z-score for \( X = 80 \): \[ Z = \frac{80 - 65}{11} = \frac{15}{11} \approx 1.364 \] Next, we will look up this Z-score in the standard normal distribution table or use a calculator to find the probability. The probability corresponding to \( Z = 1.364 \) is approximately \( 0.9147 \). So, the probability that \( X \) is less than 80 is approximately: **0.915** (rounded to the nearest thousandth).
