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Chapter 3: Problem 2

The 20 brain volumes \(\left(\mathrm{cm}^{3}\right)\) from Data Set 8 "IQ and Brain Size" in Appendix B have a mean of \(1126.0 \mathrm{~cm}^{3}\) and a standard deviation of \(124.9 \mathrm{~cm}^{3}\). Use the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of \(1440 \mathrm{~cm}^{3}\) be significantly high?

Short Answer

Expert verified
Yes, a brain volume of 1440 \(\text{cm}^{3}\) is significantly high.

Step by step solution

01

Understand the Range Rule of Thumb

The range rule of thumb states that most data will fall within two standard deviations of the mean. Specifically, we use the formulas: \[\text{Minimum usual value} = \text{Mean} - 2 \times \text{Standard Deviation}\]\[\text{Maximum usual value} = \text{Mean} + 2 \times \text{Standard Deviation}\]
02

Calculate the Minimum Usual Value

Using the formula \(\text{Minimum usual value} = \text{Mean} - 2 \times \text{Standard Deviation}\), plug in the given values:\[\text{Mean} = 1126.0 \, \text{cm}^3\]\[\text{Standard Deviation} = 124.9 \, \text{cm}^3\]\[\text{Minimum usual value} = 1126.0 - 2 \times 124.9\]\[\text{Minimum usual value} = 1126.0 - 249.8 = 876.2 \, \text{cm}^3\]
03

Calculate the Maximum Usual Value

Using the formula \(\text{Maximum usual value} = \text{Mean} + 2 \times \text{Standard Deviation}\), plug in the given values:\[\text{Mean} = 1126.0 \, \text{cm}^3\]\[\text{Standard Deviation} = 124.9 \, \text{cm}^3\]\[\text{Maximum usual value} = 1126.0 + 2 \times 124.9\]\[\text{Maximum usual value} = 1126.0 + 249.8 = 1375.8 \, \text{cm}^3\]
04

Compare the Brain Volume to the Limits

Compare the brain volume of 1440 \, \text{cm}^{3} to the calculated range:\[\text{Minimum usual value} = 876.2 \, \text{cm}^3\]\[\text{Maximum usual value} = 1375.8 \, \text{cm}^3\]Since 1440 \, \text{cm}^{3} is greater than the maximum usual value of 1375.8 \, \text{cm}^{3}, it is significantly high.

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Key Concepts

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

mean
The mean is often referred to as the average. It's a central value representing a set of numbers. To calculate the mean of a data set, sum up all the individual values, then divide by the total number of values.
For example, given the brain volume data set with a mean of 1126.0 cm^3, we already know the central point around which other values in the set are distributed.
In essence, the mean gives us an idea of the 'average brain volume' among the studied data set, providing a useful starting point for further analysis.
standard deviation
The standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
In the given problem, the standard deviation is 124.9 cm^3.
This shows how much individual brain volume measurements deviate from the average (mean) brain volume. To put it into perspective, if brain volumes were tightly clustered around the mean, the standard deviation would be smaller.
Since our standard deviation is relatively large, it tells us there鈥檚 considerable variation in the brain volumes within this data set.
significantly high
To determine if a value is significantly high or low, we use the range rule of thumb. This rule helps by setting limits derived from the mean and standard deviation.
According to the rule, most data will fall within two standard deviations of the mean.
  • Minimum usual value = Mean - 2 脳 Standard Deviation
  • Maximum usual value = Mean + 2 脳 Standard Deviation
In the example provided, we calculated:
Minimum usual value = 876.2 cm^3
Maximum usual value = 1375.8 cm^3
Any value below the minimum or above the maximum is considered significantly low or high.
For a brain volume of 1440 cm^3, which is above the maximum usual value, it is labeled as significantly high.

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Most popular questions from this chapter

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, \((b)\) median, (c) mode, (d) midrange, and then answer the given question. Listed below are the numbers of Atlantic hurricanes that occurred in each year. The data are listed in order by year, starting with the year 2000 . What important feature of the data is not revealed by any of the measures of center? $$ \begin{array}{llllllllllllll} 8 & 9 & 8 & 7 & 9 & 15 & 5 & 6 & 8 & 4 & 12 & 7 & 8 & 2 \end{array} $$

The quadratic mean (or root mean square, or R.M.S.) is used in physical applications, such as power distribution systems. The quadratic mean of a set of values is obtained by squaring each value, adding those squares, dividing the sum by the number of values \(n\), and then taking the square root of that result, as indicated below: $$\text { Quadratic mean }=\sqrt{\frac{\sum x^{2}}{n}}$$ Find the R.M.S. of these voltages measured from household current: \(0,60,110,-110,-60,0 .\) How does the result compare to the mean?

Consider a value to be significantly low if its score is less than or equal to \(-2\) or consider the value to be significantly high if its \(z\) score is greater than or equal to \(2 .\) Data Set 29 "Coin Weights" lists weights (grams) of quarters manufactured after 1964 . Those weights have a mean of \(5.63930 \mathrm{~g}\) and a standard deviation of \(0.06194 \mathrm{~g}\). Identify the weights that are significantly low or significantly high.

Consider a value to be significantly low if its score is less than or equal to \(-2\) or consider the value to be significantly high if its \(z\) score is greater than or equal to \(2 .\) In the process of designing aircraft seats, it was found that men have hip breadths with a mean of \(36.6 \mathrm{~cm}\) and a standard deviation of \(2.5 \mathrm{~cm}\) (based on anthropometric survey data from Gordon, Clauser, et al.). Identify the hip breadths of men that are significantly low or significantly high.

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are the measured radiation absorption rates (in \(\mathrm{W} / \mathrm{kg}\) ) corresponding to these cell phones: iPhone 5S, BlackBerry Z30, Sanyo Vero, Optimus V, Droid Razr, Nokia N97, Samsung Vibrant, Sony Z750a, Kyocera Kona, LG G2, and Virgin Mobile Supreme. The data are from the Federal Communications Commission. If one of each model of cell phone is measured for radiation and the results are used to find the measures of variation, are the results typical of the population of cell phones that are in use? \(\begin{array}{lllllllllll}1.18 & 1.41 & 1.49 & 1.04 & 1.45 & 0.74 & 0.89 & 1.42 & 1.45 & 0.51 & 1.38\end{array}\)
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