Questions: 11. Assume that 500,000 people take the GED exam (high school proficiency exam) each year and their scores form a normal distribution. If Danny scores two standard deviations below the mean, what percentile is he? a) 2.5 th b) 5.0 th c) 12.5 th d) 22.5 th e) Not enough information to determine.

11. Assume that 500,000 people take the GED exam (high school proficiency exam) each year and their scores form a normal distribution. If Danny scores two standard deviations below the mean, what percentile is he?
a) 2.5 th
b) 5.0 th
c) 12.5 th
d) 22.5 th
e) Not enough information to determine.

Transcript text: 11. Assume that 500,000 people take the GED exam (high school proficiency exam) each year and their scores form a normal distribution. If Danny scores two standard deviations below the mean, what percentile is he? a) 2.5 th b) 5.0 th c) 12.5 th d) 22.5 th e) Not enough information to determine.

Solution

Solution Steps

To determine Danny's percentile, we need to understand the properties of a normal distribution. A score that is two standard deviations below the mean corresponds to a specific percentile. In a standard normal distribution, this is approximately the 2.5th percentile.

Step 1: Understanding the Normal Distribution

In a normal distribution, scores are distributed symmetrically around the mean. The standard normal distribution has a mean of 0 and a standard deviation of 1.

Step 2: Calculating the Percentile

A score that is two standard deviations below the mean corresponds to a z-score of \(-2\). The cumulative distribution function (CDF) for a z-score of \(-2\) gives the percentile rank.

Step 3: Interpreting the CDF Result

The CDF value for a z-score of \(-2\) is approximately \(0.02275\), which means Danny's score is at the 2.275th percentile.