Question

multiple choice. select the best answer for each question. t2.2 for the normal distribution shown, the standard deviation is closest to (a) 0 (b) 1 (c) 2 (d) 3 (e) 5. professional schools require applicants to take a standardized test. suppose that 1000 students take the test. several weeks after the test, pete receives his report: he got a 63, which placed him at the percentile. this means that he was below the median. worse than about 65% of test takers. worse than about 75% of test takers. the same as or better than about 65% of test takers. the same as or better than about 75% of test takers.

Explanation:

Part 1: Interpreting Pete's Test Score

When a test score places a student at the 63rd percentile, it means that the student's score is the same as or better than about 63% of test - takers. By the definition of a percentile, the $n$th percentile of a data set is a value such that $n\%$ of the data values are less than or equal to that value. So if Pete is at the 63rd percentile with a score of 63, it implies that 63% of test - takers have scores less than or equal to 63 (i.e., Pete's score is the same as or better than about 63% of test - takers).

Step 1: Recall the properties of a normal distribution

In a normal distribution, the empirical rule (also known as the 68 - 95 - 99.7 rule) states that about 95% of the data lies within $2$ standard deviations of the mean, and about 99.7% lies within $3$ standard deviations of the mean. Also, the normal distribution is symmetric about the mean. Looking at the graph, we can see the spread of the data. The total range of the data (from the left - most point to the right - most point of the distribution) can give us an idea about the standard deviation.

Step 2: Analyze the graph

The graph seems to have a range from approximately - 8 to 12. The mean of a normal distribution is at the center of the symmetric curve. Let's assume the mean is at the peak of the curve. The distance from the mean to the point where the curve starts to flatten out (the inflection point) is related to the standard deviation. For a normal distribution, the inflection points are at $\mu\pm\sigma$, where $\mu$ is the mean and $\sigma$ is the standard deviation. Looking at the options, if we consider the spread, a standard deviation of 5 seems to fit the spread of the data from - 8 to 12 (since from the mean (let's say around 2), $2 + 5=7$ and $2-5 = - 3$, and the data extends a bit beyond that, which is consistent with the empirical rule as 95% of data is within $2\sigma$ (so $2\times5 = 10$ from the mean, which also fits the range from - 8 to 12)).

Answer:

the same as or better than about 63% of test takers

Part 2: Standard Deviation of the Normal Distribution