Question

a student with a score of 90 on the above test would be in what percentile? * 1 point
3rd
84th
90th
97th
which of the following scores would be within one standard deviation of the mean? * 1 point
96%
65%
81%
52%
the teacher who gave the test then realizes that she made a scoring error, which she corrects by adding 5 points to each student’s score. the standard deviation of the students’ new scores is * 1 point
10
2.5

Explanation:

Step1: Recall properties of normal - distribution and standard - deviation

In a normal distribution, approximately 68% of the data lies within one standard deviation of the mean. The scores within one standard deviation of the mean are symmetrically distributed around the mean. However, since no mean and standard - deviation values are given for the first two questions, we assume we are working with general knowledge of normal - distribution concepts. For the third question, recall the property of standard deviation when a constant is added to each data point.

Step2: Analyze the percentile question

Percentile represents the percentage of scores that a particular score is greater than or equal to. Without more context about the distribution of scores, we assume a normal - distribution for general understanding. But since we are just validating an answer, we note that the 97th percentile means the student has performed better than 97% of the students.

Step3: Analyze the standard - deviation range question

In a normal distribution, scores within one standard deviation of the mean cover a significant portion of the data. We assume we are looking for a typical value within this range. Since no specific mean and standard - deviation are given, we rely on the fact that about 68% of data is within one standard deviation of the mean.

Step4: Analyze the effect of adding a constant on standard deviation

The formula for the standard deviation of a set of data \(x_1,x_2,\cdots,x_n\) is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\). If we add a constant \(c\) to each data point, so the new data points are \(y_i=x_i + c\), the new mean \(\bar{y}=\bar{x}+c\). Then \(y_i-\bar{y}=(x_i + c)-(\bar{x}+c)=x_i-\bar{x}\). So the standard deviation of the new data set is the same as the standard deviation of the original data set. Since no original standard - deviation value is given, we just know the property that adding a constant to each data point does not change the standard deviation.

Answer:

1. A student with a score of 90 on the above test would be in what percentile? - The given answer is 97th.
2. Which of the following scores would be within one standard deviation of the mean? - The given answer is 52%.
3. The teacher who gave the test then realizes that she made a scoring error, which she corrects by adding 5 points to each student’s score. The standard deviation of the students’ new scores is - There is not enough information given in the problem to determine a numerical value for the standard deviation, but we know that adding a constant to each data - point does not change the standard deviation. So, if we assume the original standard deviation was not given, we can't choose 10 or 2.5 based on the information provided. But conceptually, the standard deviation remains the same as the original (unknown) standard deviation.