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Chapter 3: Problem 15

Use z scores to compare the given values. Birth Weights Based on Data Set 4 "Births" in Appendix B, newborn males have weights with a mean of \(3272.8 \mathrm{g}\) and a standard deviation of \(660.2 \mathrm{g}\). Newborn females have weights with a mean of \(3037.1 \mathrm{g}\) and a standard deviation of \(706.3 \mathrm{g}\). Who has the weight that is more extreme relative to the group from which they came: a male who weighs \(1500 \mathrm{g}\) or a female who weighs \(1500 \mathrm{g} ?\)

Short Answer

Expert verified
The male weighing 1500 g has a more extreme weight relative to his group.

Step by step solution

01

Understand the problem

The goal is to use z-scores to determine which newborn has a weight that is more extreme relative to their respective group鈥攁 male or a female both weighing 1500 g.
02

Recall the z-score formula

The z-score formula is given by \[ z = \frac{X - \mu}{\sigma} \] where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
03

Compute the z-score for the male newborn

For the male: \( X = 1500 \, \text{g}, \mu = 3272.8 \, \text{g}, \sigma = 660.2 \, \text{g} \). Substitute these values into the z-score formula: \[ z_{male} = \frac{1500 - 3272.8}{660.2} \approx \frac{-1772.8}{660.2} \approx -2.68 \]
04

Compute the z-score for the female newborn

For the female: \( X = 1500 \, \text{g}, \mu = 3037.1 \, \text{g}, \sigma = 706.3 \, \text{g} \). Substitute these values into the z-score formula: \[ z_{female} = \frac{1500 - 3037.1}{706.3} \approx \frac{-1537.1}{706.3} \approx -2.18 \]
05

Compare the z-scores

The male has a z-score of approximately \(-2.68\), and the female has a z-score of approximately \(-2.18\). Since z-scores represent the number of standard deviations away from the mean, a more negative z-score indicates a value that is more extreme relative to the group.
06

Conclusion

The male newborn's weight is more extreme relative to the group, as \(-2.68\) is more extreme than \(-2.18\).

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Key Concepts

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

Birth Weight Analysis
Birth weight is an essential parameter in neonatal health. It helps evaluate whether a newborn is underweight, healthy, or overweight. In this exercise, we focus on comparing the weights of newborn male and female babies. By using statistical methods like z-scores, we can determine how extreme a specific birth weight is compared to the average weights of their respective groups.
Newborn males have an average weight of 3272.8 grams, and newborn females have an average of 3037.1 grams. Understanding these average weights and their standard deviations provides a foundation for further analysis using z-scores.
Z-Score Calculation
A z-score measures how many standard deviations a data point is from the mean. It helps in understanding if a value is typical or unusual compared to a dataset. The formula for calculating the z-score is:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
  • \(X\) is the data point (birth weight in this case)
  • \(\mu\) is the mean
  • \(\sigma\) is the standard deviation
By substituting the necessary values, we can compute the z-scores for both male and female newborn weights of 1500 grams.
For males:
\( z_{male} = \frac{1500 - 3272.8}{660.2} \approx -2.68 \)
For females:
\( z_{female} = \frac{1500 - 3037.1}{706.3} \approx -2.18 \)
Standard Deviation Interpretation
Standard deviation is a measure of 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 our context:
  • The standard deviation for newborn males is 660.2 grams.
  • The standard deviation for newborn females is 706.3 grams.
These figures help us understand the spread of birth weights in each group. A greater standard deviation means more variability in the birth weights. This variability is critical when interpreting and comparing z-scores.
Statistical Comparison
To determine which newborn weight is more extreme relative to its respective group, we compare their z-scores. A more negative z-score indicates a value that is further from the mean, hence more unusual.
In this exercise:
  • The male newborn has a z-score of approximately -2.68.
  • The female newborn has a z-score of approximately -2.18.
Since -2.68 is more negative than -2.18, the male newborn's weight is more extreme relative to his group compared to the female newborn's weight. Thus, the male baby weighing 1500 grams is further from the average male birth weight than the female baby weighing 1500 grams is from the average female birth weight.

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Most popular questions from this chapter

Express all z scores with two decimal places. For the Verizon airport data speeds (Mbps) listed in Data Set 32 "Airport Data Speeds" in Appendix B, the highest speed of 77.8 Mbps was measured at Atlanta's (ATL) international airport. The complete list of 50 Verizon data speeds has a mean of \(\bar{x}=17.60\) Mbps and a standard deviation of \(s=16.02 \mathrm{Mbps.}\) a. What is the difference between Verizon's data speed at Atlanta's international airport and the mean of all of Verizon's data speeds? b. How many standard deviations is that [the difference found in part (a)]? c. Convert Verizon's data speed at Atlanta's international airport to a \(z\) score. d. If we consider data speeds that convert to \(z\) scores between -2 and 2 to be neither significantly low nor significantly high, is Verizon's speed at Atllanta significant?

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: $$\text { Quadratic mean }=\sqrt{\frac{\Sigma x^{2}}{n}}$$ 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?

Find the coefficient of variation for each of the two samples; then compare the variation. (The same data were used in Section 3-I.) Listed below are pulse rates (beats per minute) from samples of adult males and females (from Data Set 1 "Body Data" in Appendix B). Does there appear to be a difference? $$\begin{array}{llllllllllllllll} \text { Male: } & 86 & 72 & 64 & 72 & 72 & 54 & 66 & 56 & 80 & 72 & 64 & 64 & 96 & 58 & 66 \\ \text { Female: } & 64 & 84 & 82 & 70 & 74 & 86 & 90 & 88 & 90 & 90 & 94 & 68 & 90 & 82 & 80 \end{array}$$

Find the mean and median for each of the two samples, then compare the two sets of results. Listed below are pulse rates (beats per minute) from samples of adult males and females (from Data Set 1 "Body Data" in Appendix B). Does there appear to be a difference? $$\begin{array}{llllllllllll}\text { Male: } & 86 & 72 & 64 & 72 & 72 & 54 & 66 & 56 & 80 & 72 & 64 & 64 & 96 & 58 & 66 \\\\\text { Female: } & 64 & 84 & 82 & 70 & 74 & 86 & 90 & 88 & 90 & 90 & 94 & 68 & 90 & 82 & 80\end{array}$$

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 annual costs (dollars) of tuition and fees at the 10 most expensive colleges in the United States for a recent year (based on data from U.S. News \& World Report). The colleges listed in order are Columbia, Vassar, Trinity, George Washington, Carnegie Mellon, Wesleyan, Tulane, Bucknell, Oberlin, and Union. What does this "top \(10^{\circ}\) list tell us about those costs for the population of all U.S. college tuitions? $$\begin{array}{rl}49,138 & 47,890 \quad 47,510 \quad 47,343 \quad 46,962 \quad 46,944 \quad 46,930 \quad 46,902 \quad 46,870 \quad 46,785\end{array}$$
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