Question

question 8 1 pts reggie conducts research and wants to know the probability of being in the top 65% of people in his normally distributed sample. his mean is 85 with a standard deviation of 12. which of the following is not true regarding reggies sample? an individual in the top 65% may have a negative z - score value 65% roughly translates to a z - score of +0.39 an individual in the top 65% will only have a positive z - score value someone with a z - score of +1.00 will be in the top 65%

Explanation:

Step1: Understand z - score and normal distribution

In a normal distribution, the z - score is calculated as $z=\frac{x - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation. The area under the normal - distribution curve represents probabilities.

Step2: Analyze the top 65%

If we want the top 65% of the data, we are looking at the area to the right of a certain z - score. The area to the left of this z - score is $1 - 0.65=0.35$. Looking up 0.35 in the standard normal distribution table (z - table), the z - score corresponding to an area of 0.35 is approximately $z\approx - 0.39$. So, an individual in the top 65% can have a negative z - score.

Step3: Evaluate each option

• Option 1: An individual in the top 65% may have a negative z - score value. This is true as we found the z - score corresponding to the area of 0.35 (left - hand side of the value for top 65%) is negative.
• Option 2: 65% roughly translates to a z - score of + 0.39. This is false. As we calculated, the z - score for the area corresponding to the non - top 65% (area to the left) is approximately $z\approx - 0.39$.
• Option 3: An individual in the top 65% will only have a positive z - score value. This is false because we've shown that the z - score can be negative.
• Option 4: A z - score of + 1.00 corresponds to an area to the left of approximately 0.8413 in the standard normal distribution. So the area to the right (top) is $1 - 0.8413 = 0.1587$, which means a z - score of + 1.00 is not in the top 65%. This is false.

Answer:

The statement "65% roughly translates to a z - score of + 0.39" is not true.