Problem

5. 3pts Find the $z_{1}$ and $z_{2}$, such that they separate the middle $68 \%$ of the distribution from the rest. Draw a bell-shaped curve, label your mean and shade the area that you are trying to find.

Answer

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Answer

\(\boxed{z_{1} = -1, z_{2} = 1}\)

Steps

Step 1 :The problem is asking for the z-scores that correspond to the middle 68% of a normal distribution. In a standard normal distribution, the middle 68% of data falls between -1 and 1 standard deviations from the mean.

Step 2 :Therefore, the z-scores that separate the middle 68% of the distribution from the rest are \(z_{1} = -1\) and \(z_{2} = 1\).

Step 3 :The negative z-score represents the point below the mean where 16% of the data falls, and the positive z-score represents the point above the mean where 16% of the data falls.

Step 4 :Therefore, the middle 68% of the data falls between these two z-scores.

Step 5 :\(\boxed{z_{1} = -1, z_{2} = 1}\)

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