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Chapter 7: Problem 38

Suppose a set of test scores is approximately bell-shaped, with a mean of 70 and a range of 50 . Approximately, what would the minimum and maximum test scores be?

Short Answer

Expert verified
Minimum: 45, Maximum: 95.

Step by step solution

01

Understand the Problem

We are given a bell-shaped (normal distribution) set of test scores with a mean of 70 and a range of 50. We are tasked with finding the approximate minimum and maximum test scores.
02

Determine the Range Relationship

The range of a distribution is the difference between the maximum and minimum values. We can set up the equation for the range: \( \text{Range} = \text{Maximum} - \text{Minimum} \). Here, the range is given as 50.
03

Set Up the Equations

If we assume the data is symmetrically distributed around the mean, the minimum and maximum can be represented with the mean in the center. So, \( \text{Maximum} = 70 + \frac{50}{2} \) and \( \text{Minimum} = 70 - \frac{50}{2} \).
04

Calculate the Maximum Score

Calculate the maximum score as \( \text{Maximum} = 70 + \frac{50}{2} = 70 + 25 = 95 \).
05

Calculate the Minimum Score

Calculate the minimum score as \( \text{Minimum} = 70 - \frac{50}{2} = 70 - 25 = 45 \).
06

Conclusion

The approximate minimum test score is 45, and the maximum test score is 95.

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Key Concepts

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

Mean
The mean is a central measure in statistics that represents the average value within a data set. To calculate the mean, sum all the numbers in the data set and then divide by the total number of numbers. The mean is important because it provides a single value that summarizes the essence of a data set.
  • It's also particularly useful when a data set is symmetric or normally distributed, as the mean will align with the peak of the data's "bell curve."
  • In the context of the original problem, a mean of 70 suggests that the test scores on average center around this value.
Understanding the mean helps us interpret the distribution and variability of data, making it a foundational concept in statistics.
Range in Statistics
In statistics, the range is a measure of how spread out the values in a data set are. It is calculated by subtracting the smallest value from the largest value. The range gives insight into the variability of the data but does not provide information about the distribution's shape.
  • In the problem, with a range of 50, this tells us that the difference between the highest and lowest test scores is 50 points, informing us about the span of the scores.
  • Knowing the range helps in understanding how dispersed the scores are, which is crucial for evaluating the consistency and reliability of the data.
While the range provides valuable information, it's often used in conjunction with other measures, like variance or standard deviation, for a fuller picture of data variability.
Symmetrical Distribution
A symmetrical distribution is one where the left and right sides of the distribution are mirror images of each other, centering around a particular point, often the mean. In such distributions, data is evenly distributed on either side of the mean, forming a normal distribution or bell curve.
  • In this scenario, the test scores have a symmetrical (bell-shaped) distribution with a mean of 70, indicating a normal distribution.
  • This symmetry helps in simplifying calculations and predictions, as the mean value is at the center, and scores fall-off evenly on either side.
A key property of a symmetrical distribution is that it allows for the use of the mean as a representative measure of central tendency, making it easier to determine values like the minimum and maximum scores around the mean.

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Most popular questions from this chapter

Give an example of a measurement for which the mode would be more useful than the median or the mean as an indicator of the "typical" value. (Do not reuse the example of kindergarten children's ages.)

Suppose you are interested in genealogy, and want to try to predict your potential longevity by using the ages at death of your ancestors. You find out the ages at death for the eight greatgrandparents of your mother and your father. Suppose the ages (in numerical order) are as follows: Mother's Great-Grandparents: $$78,80,80,81,81,82,82,84$$ Father's Great-Grandparents: $$30,50,77,80,82,90,95,98$$ a. Compare the medians for the two sets of ages. How do they compare? b. Compare the means for the two sets of ages. How do they compare?' c. Find the standard deviation for each set of ages. d. Which set do you think is more useful for predicting longevity in your family? Explain.

In each of the following cases, would the mean or the median probably be higher, or would they be about equal? a. Salaries in a company employing 100 factory workers and two highly paid executives. b. Ages at which residents of a suburban city die, including everything from infant deaths to the most elderly.

a. Give an example of a set of five numbers with a standard deviation of 0 . b. Give an example of a set of four numbers with a mean of 15 and a standard deviation of 0 . c. Is there more than one possible set of numbers that could be used to answer part (a)? Is there more than one possible set of numbers that could be used to answer part (b)? Explain.

Construct an example, and draw a histogram for a measurement that you think would be skewed to the right.
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