Question

Assume the random variable X is normally distributed with mean μ=50 and standard deviation 0=7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded P(x>38)

Answer 1

The probability that the random variable
`\( X \)` is greater than 38, denoted as
`\( P(X > 38) \),` in a normal distribution with mean
`\( \mu = 50 \)` and standard deviation
`\( \sigma = 7 \)` is approximately 0.9826.

Step-by-step explanation:

To calculate the probability
`\( P(X > 38) \)`, we need to standardize the value 38 using the standard normal distribution. The standard normal distribution has a mean
`(\( \mu \))` of 0 and a standard deviation
`(\( \sigma \))` of 1. The formula for standardizing a value
`\( x \)` from a normal distribution with mean
`\( \mu \)` and standard deviation
`\( \sigma \)` is given by
`\( z = \frac{{x - \mu}}{{\sigma}} \).`

For this problem, the standardized value
`\( z \)` is calculated as
`\( z = \frac{{38 - 50}}{{7}} = -1.7143 \).` Using a standard normal distribution table or calculator, we find that the probability corresponding to
`\( z = -1.7143 \)` is approximately 0.0414. Since we are interested in
`\( P(X > 38) \),` we subtract this probability from 1 to get
`\( P(X > 38) \approx 1 - 0.0414 = 0.9586 \).`

Thus, the probability that the random variable
`\( X \)` is greater than 38 is approximately 0.9586 or 95.86%. To visually represent this probability, you can draw a normal curve with the shaded area to the right of 38, illustrating the portion of the distribution corresponding to
`\( P(X > 38) \).`Understanding how to interpret probabilities in the context of a normal distribution is crucial in statistics and data analysis.