Question

Assuming the random variable X follows a normal distribution with a mean μ = 50 and standard deviation σ = 7, calculate the probability. Ensure to plot a normal curve where the area corresponding to the probability is shaded: P(35 < X < 65).

Answer 1

To calculate P(35 < X < 65) for a normal distribution with mean μ = 50 and standard deviation σ = 7, transform X values to z-scores and use a normal distribution table or calculator to find the probability for each z-score, then determine the area between them which represents the desired probability.

Step-by-step explanation:

To calculate the probability P(35 < X < 65) for a normal distribution with mean μ = 50 and standard deviation σ = 7, we need to standardize these values and use a normal distribution table, a computer, or a calculator. First, we calculate the z-scores corresponding to X = 35 and X = 65:

• For X = 35: z = (35 - 50) / 7 = -15 / 7 ≈ -2.14
• For X = 65: z = (65 - 50) / 7 = 15 / 7 ≈ 2.14

We then find the probability for each z-value using the standard normal distribution and subtract the smaller from the larger to find the shading area between. In this case, the probability that X falls between 35 and 65 is roughly the composite area under the standard normal curve between z-scores -2.14 and 2.14.

Typically, values within 2 standard deviations from the mean in a normal distribution contain about 95% of the data. So, we would expect P(35 < X < 65) to be close to this percentage.