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Chapter 2: Problem 137

Which \(x\) value has the higher position relative to the set of data from which it comes? \(\mathrm{A}: x=85,\) where mean \(=72\) and standard deviation \(=8\) \(\mathbf{B}: x=93,\) where mean \(=87\) and standard deviation \(=5\)

Short Answer

Expert verified
In terms of position relative to their respective data sets, \(x = 85\) in Set A is higher because it has a higher z-score.

Step by step solution

01

Calculate the Z-score for Set A

To calculate the z-score, the formula is: \( Z = (x - \mu) / \sigma \) where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. Calculating for Set A gives: \( Z = (85 - 72) / 8 = 1.625 \)
02

Calculate the Z-score for Set B

Applying the same formula for Set B: \( Z = (93 - 87) / 5 = 1.2 \)
03

Compare the results

The value from Set A has a higher position compared to its own data set since its z-score is higher than that from Set B.

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Key Concepts

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

Understanding Standard Deviation
Standard deviation is a measure of how spread out the values in a data set are. It tells us how much the individual data points differ from the mean of the set.
  • If the standard deviation is large, it means the data points are spread out over a wide range of values.
  • If the standard deviation is small, it indicates that the data points are closely clustered around the mean.
This is crucial when calculating the Z-score, as it directly influences how much a given value deviates from the mean relative to the entire data set. For example, in Set A, the standard deviation is 8. This indicates more variability within the data set compared to Set B, where the standard deviation is 5. Understanding this difference helps in determining how unusual or typical a particular value is in its respective data set.
Calculating the Mean
The mean, often referred to as the average, is a central value for a set of numbers. It's calculated by summing all numbers in a data set and then dividing by the count of those numbers.
  • In Set A, the mean is 72, serving as an anchor point for assessing how individual values compare within the data set.
  • For Set B, the mean is 87, similarly serving as a point of comparison.
When we calculate a Z-score, we measure how far a value is from the mean in terms of standard deviations. This means the positioning of the mean is pivotal in understanding the relative positioning of an individual data point, like the 85 in Set A and the 93 in Set B.
Data Sets Comparison
When comparing data sets, it's important to evaluate each in relation to its own mean and standard deviation. This is where Z-scores become essential.
  • A Z-score tells us how many standard deviations a value is from the mean. A higher Z-score indicates that the value is further away from the mean relative to the spread of the data set.
  • In the given exercise, the Z-score of Set A's value (85) is 1.625, which is higher than the Z-score of Set B's value (93), which is 1.2. This suggests that the value 85 in Set A is more unusually high compared to its data set, than the 93 is in Set B.
Thus, comparing Z-scores allows us to see which value holds a more prominent position within its context, helping to better understand the relative standing of values across different data sets.

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Most popular questions from this chapter

On the first day of class last semester, 50 students were asked for the one- way distance from home to college (to the nearest mile). The resulting data follow: $$\begin{array}{rrrrrrrrrr} \hline 6 & 5 & 3 & 24 & 15 & 15 & 6 & 2 & 1 & 3 \\ 5 & 10 & 9 & 21 & 8 & 10 & 9 & 14 & 16 & 16 \\ 10 & 21 & 20 & 15 & 9 & 4 & 12 & 27 & 10 & 10 \\ 3 & 9 & 17 & 6 & 11 & 10 & 12 & 5 & 7 & 11 \\ 5 & 8 & 22 & 20 & 13 & 7 & 8 & 13 & 4 & 18 \\ \hline \end{array}$$ a. Construct a grouped frequency distribution of the data by using \(1-4\) as the first class. b. Calculate the mean and the standard deviation. c. Determine the values of \(\bar{x} \pm 2 s\), and determine the percentage of data within 2 standard deviations of the mean.

What is the mean weekly pay if 5 employees earn \(\$ 425\) per week, 3 earn \(\$ 750\) per week, and 1 earns \(\$ 1340 ?\)

Which \(x\) value has the lower position relative to the set of data from which it comes? \( \mathrm{A}: x=28.1, \text { where } \bar{x}=25.7 \text { and } s=1.8 \) \(\mathbf{B}: x=39.2,\) where \(\bar{x}=34.1\) and \(s=4.3\)

According to the empirical rule, almost all the data should lie between \((\bar{x}-3 s)\) and \((\bar{x}+3 s) .\) The range accounts for all the data. a. What relationship should hold (approximately) between the standard deviation and the range? b. How can you use the results of part a to estimate the standard deviation in situations when the range is known?

A sample has a mean of 50 and a standard deviation of \(4.0 .\) Find the \(z\) -score for each value of \(x\) : a. \(x=54\) b. \(x=50\) c. \(x=59\) d. \(x=45\)
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