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Solution Steps

The image provides the following information:

• The random variable _x_ is normally distributed.
• The mean (µ) is 50.
• The standard deviation (σ) is 7.
• The question asks for $P(36 < x < 62)$.
• It also asks which of the following normal curves corresponds to $P(36 < x < 62)$.

To find the correct normal curve, we need to calculate the z-scores for x = 36 and x = 62 using the formula: $z = \frac{x - µ}{σ}$.

$z_{36} = \frac{36 - 50}{7} = \frac{-14}{7} = -2$

$z_{62} = \frac{62 - 50}{7} = \frac{12}{7} ≈ 1.71$

The z-scores tell us how many standard deviations each value is from the mean. $x = 36$ is 2 standard deviations _below_ the mean, and $x = 62$ is approximately 1.71 standard deviations _above_ the mean.

The correct normal curve should have the shaded region between approximately -2 and 1.71 standard deviations from the mean. Curve A corresponds to the area between -1 standard deviation and 1 standard deviation around the mean (50). This doesn't match our z-scores. Curve B shades the area between approximately -2 and approximately 1.7 standard deviations.

Final Answer: