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Chapter 5: Problem 1

The mean score on the Stats exam was 75 points with a standard deviation of 5 points, and Gregor's z-score was -2 . How many points did he score?

Short Answer

Expert verified
Gregor scored 65 points on the Stats exam.

Step by step solution

01

- Understanding the Z-score

Z-score is the number of standard deviations from the mean a data point is. In the case of Gregor, a z-score of -2 means his score was two standard deviations below the mean.
02

- Standard Deviation Calculation

Multiply the z-score by the standard deviation. The standard deviation is given as 5 points. Hence, -2 (Gregor's z-score) multiplied by 5 (standard deviation) equals -10.
03

- Add Mean Score

Add the product of the standard deviation and the z-score to the mean. The mean score is given as 75 points. Hence, 75 (mean score) plus -10 (product of standard deviation and z-score) equals 65.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation
When working with data, understanding how it spreads out from the average value is crucial. This is where the concept of standard deviation comes into play. In simple terms, the standard deviation tells us how much the different values deviate from the mean (average).
For example, if the scores are clustered closely around the mean, the standard deviation will be small. Conversely, if the scores are spread out widely, the standard deviation will be larger.
In the problem, the standard deviation of the scores is given as 5 points. This figure helps us understand how much variation we can expect from the average score, which is 75 points in this case.
  • Small standard deviation: Scores are close to the mean.
  • Large standard deviation: Scores are spread out from the mean.
Mean
The mean, often referred to as the average, is a central value of a set of numbers. To calculate it, you sum up all the numbers and divide by the total count of numbers.
The mean gives us a single value that represents the entire dataset, making it easier to understand and analyze. In our context, the mean score of the Stats exam is 75 points. This means that, on average, students scored 75 points in the exam.
  • Add up all the scores.
  • Divide by the total number of scores.
This mean score serves as a baseline to determine how each individual performed in relation to others.
Score Calculation
Score calculation using the z-score involves understanding how far and in which direction a particular score deviates from the mean.
A z-score is a statistical measurement that indicates how many standard deviations a data point is from the mean. It can be positive or negative, indicating whether the data point is above or below the mean, respectively.
Here鈥檚 how to calculate the score from a given z-score:
  • First, multiply the z-score by the standard deviation.
  • In our scenario, Gregor's z-score is -2, and when multiplied by the standard deviation of 5, gives us -10.
  • Next, adjust the mean by this product. Subtract -10 from the mean score of 75 points, resulting in 65 points.
  • Thus, Gregor's actual score is 65 points.
This calculation helps us determine quantitatively how Gregor鈥檚 performance compared to the class average.

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