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Chapter 6: Problem 7

Suppose your Statistics professor reports test grades as z-scores, and you got a score of \(2.20\) on an exam. Write a sentence explaining what that means.

Short Answer

Expert verified
Your score was 2.20 standard deviations above the mean.

Step by step solution

01

Understanding Z-Scores

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean.
02

Interpreting the Z-Score

You received a z-score of 2.20, which means your score is 2.20 standard deviations above the mean score of the exam.
03

Implications of the Z-Score

A z-score of 2.20 indicates that your performance on the test was higher than the average performance of your peers, as you scored significantly above the mean.

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Key Concepts

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

Standard Deviation
When you hear the term 'standard deviation' in statistics, it's all about the spread of a dataset. Specifically, it tells you how much individual data points in a group vary from the mean. The mean is simply the average of all the scores, calculated by adding them up and dividing by the number of scores. The standard deviation gives a kind of average distance from that middle point.

  • A low standard deviation means the data points are close to the mean.
  • A high standard deviation suggests a wider range of values.
Understanding standard deviation helps you grasp how diverse or consistent a data set is, such as test scores in a class.
Mean
The mean is one of the most fundamental concepts in statistics. Essentially, the mean is the average of a set of numbers. To find it, you add up all the numbers and then divide by how many numbers you added together.

The mean provides a central value around which all other values in the dataset are distributed. It's essential in calculating the z-score because the z-score is determined by comparing each value to the mean in terms of standard deviations. A high z-score indicates how far above or below the average a score is.
  • To calculate the mean: Sum all values.
  • Divide by the total number of observations.
This helps in offering a point of reference for interpreting data values.
Statistics
Statistics is a field dealing with data collection, analysis, interpretation, and presentation. It's all about making sense of data and figuring out patterns or trends. Statistics use various tools like mean, median, mode, variance, and standard deviation to provide insights into data.

The core idea of statistics is to take a massive amount of information and simplify it into understandable results. For example, in education, statistics help summarize scores on tests, like our example with z-scores. A great way to think about statistics is it serves as the map that guides us through the rough terrain of numbers, making sense of what might initially appear to be random figures.
Test Scores
Test scores are numerical data that represent a student's performance on an exam. They can be used to assess an individual's understanding of a subject or compare performance across a period.

In terms of statistics, these scores are often analyzed to determine the effectiveness of teaching, identify strengths and weaknesses, or evaluate student progress. The concept of z-scores plays a significant role here:
  • Z-score shows how a single score compares to the mean.
  • It is expressed in terms of the number of standard deviations from the mean.
A z-score of 2.20, for instance, means a student's score is well above the average, indicating strong performance compared to peers. Through z-scores, educators can easily spot exceptional or underperforming results.

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Most popular questions from this chapter

In Exercise 17 we suggested the model \(N(1152,84)\) for weights in pounds of yearling Angus steers. What weight would you consider to be unusually low for such an animal? Explain.

NFL data from the 2006 football season reported the number of yards gained by each of the league's 167 wide receivers: a) According to the Normal model, what percent of receivers would you expect to gain fewer yards than 2 standard deviations below the mean number of yards? b) For these data, what does that mean? c) Explain the problem in using a Normal model here.

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?

In the Normal model \(N(100,16)\), what cutoff value bounds a) the highest \(5 \%\) of all IQs? b) the lowest \(30 \%\) of the IQs? c) the middle \(80 \%\) of the IQs?

The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. a) How many standard deviations from the mean would a steer weighing 1000 pounds be? b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds?
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