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Chapter 3: Problem 11

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.

Short Answer

Expert verified
Significantly low weights are <= 5.51542 g, and significantly high weights are >= 5.76318 g.

Step by step solution

01

Understand the problem

Identify the given values and understand the thresholds for significantly low and high Z-scores. Given values: mean = 5.63930 g, standard deviation = 0.06194 g. Thresholds: Z <= -2 (significantly low), Z >= 2 (significantly high).
02

Recall the Z-score formula

The formula for the Z-score is given by: \[ Z = \frac{X - \mu}{\sigma} \]Where X is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
03

Set up equations for threshold values

To find the significantly low and high weights, solve for X in the Z-score formula using Z = -2 and Z = 2:\[ -2 = \frac{X - 5.63930}{0.06194} \] \[ 2 = \frac{X - 5.63930}{0.06194} \]
04

Solve for low threshold weight (Z = -2)

Multiply both sides of the equation \( -2 = \frac{X - 5.63930}{0.06194} \) by 0.06194:\[ -2 * 0.06194 = X - 5.63930 \]\[ -0.12388 = X - 5.63930 \] Add 5.63930 to both sides:\[ X = 5.63930 - 0.12388 \]\[ X = 5.51542 \text{ g} \]
05

Solve for high threshold weight (Z = 2)

Multiply both sides of the equation \( 2 = \frac{X - 5.63930}{0.06194} \) by 0.06194:\[ 2 * 0.06194 = X - 5.63930 \]\[ 0.12388 = X - 5.63930 \] Add 5.63930 to both sides:\[ X = 5.63930 + 0.12388 \]\[ X = 5.76318 \text{ g} \]
06

Conclusion

Weights significantly low are <= 5.51542 g. Weights significantly high are >= 5.76318 g.

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Key Concepts

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

Normal Distribution
The normal distribution is a widely used statistical concept characterized by its bell-shaped curve. This curve is symmetrical, meaning the left and right sides are mirror images of each other. Each side of the curve extends indefinitely, getting closer but never touching the horizontal axis. The highest point in the curve represents the mean. To understand it better:
  • The normal distribution is used to represent real-valued random variables.
  • Most occurrences take place near the mean, dwindling as they move away.
  • In practical terms, weights of manufactured quarters can follow a normal distribution to assess consistency.
Grasping the normal distribution is fundamental in statistics as it forms the basis for calculating probabilities and making inferences about a population.
Significance Thresholds
Significance thresholds are critical in determining which data points hold statistical importance. In the context of Z-scores, a threshold helps us decide if a value is significantly high or low based on its distance from the mean.
  • Z-scores below -2 are *significantly low*; they deviate more than two standard deviations below the mean.
  • Z-scores above 2 are *significantly high*; they deviate more than two standard deviations above the mean.
This threshold gives us a clear criterion for identifying unusual data points. When you calculated that weights 鈮� 5.51542 g were significantly low and 鈮� 5.76318 g were significantly high, you used this concept. Understanding significance thresholds helps in making informed decisions in various fields, from quality control to research hypothesis testing.
Standard Deviation
Standard deviation (蟽) measures the amount of variation or dispersion of a set of values. A smaller standard deviation means that values tend to be close to the mean. Larger standard deviation indicates that values are more spread out. Here's how it鈥檚 relevant:
  • Standard deviation allows us to understand how much individual weights of quarters vary from the average weight.
  • It is central to calculating Z-scores, which are key to identifying outliers.
The given standard deviation of 0.06194 g for the weights indicates how tightly clustered the weights are around the mean. It's crucial in understanding whether a particular quarter's weight is typical or an outlier.
Mean Calculation
The mean (渭) is the average of a set of numbers. Calculating the mean involves adding up all data points and then dividing by the number of points. Here鈥檚 a step-by-step for understanding:
  • Sum all observed weights of quarters.
  • Divide by the total number of weights to get the mean (e.g., given mean = 5.63930 g).
Knowing the mean is essential because it provides a central value from which deviations are measured. The mean helps in calculating Z-scores, which identify how far a data point is from the average in terms of standard deviations. Understanding mean and its role in statistical analysis simplifies many complex calculations.

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Most popular questions from this chapter

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? \(\begin{array}{llllllllllllllll}\text { Male: } & 86 & 72 & 64 & 72 & 72 & 54 & 66 & 56 & 80 & 72 & 64 & 64 & 96 & 58 & 66 \\ \text { Female: } & 64 & 84 & 82 & 70 & 74 & 86 & 90 & 88 & 90 & 90 & 94 & 68 & 90 & 82 & 80\end{array}\)

Use \(z\) scores to compare the given values. 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 \(1500 \mathrm{~g}\) ?

A student of the author earned grades of \(\mathrm{A}, \mathrm{C}, \mathrm{B}, \mathrm{A}\), and \(\mathrm{D}\). Those courses had these corresponding numbers of credit hours: \(3,3,3,4\), and \(1 .\) The grading system assigns quality points to letter grades as follows: \(\mathrm{A}=4 ; \mathrm{B}=3 ; \mathrm{C}=2 ; \mathrm{D}=1 ; \mathrm{F}=0\). Compute the grade-point average (GPA) and round the result with two decimal places. If the dean's list requires a GPA of \(3.00\) or greater, did this student make the dean's list?

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 prices in dollars for one night at different hotels located on Las Vegas Boulevard (the "Strip"). If you decide to stay at one of these hotels, what statistic is most relevant, other than the measures of center? Apart from price, identify one other important factor that would affect your choice. $$ \begin{array}{llllllll} 212 & 77 & 121 & 104 & 153 & 264 & 195 & 244 \end{array} $$

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 highest amounts of net worth (in millions of dollars) of celebrities. The celebrities are Tom Cruise, Will Smith, Robert De Niro, Drew Carey, George Clooney, John Travolta, Samuel L. Jackson, Larry King, Demi Moore, and Bruce Willis. Are the measures of variation typical for all celebrities? $$ \begin{array}{llllllllll} 250 & 200 & 185 & 165 & 160 & 160 & 150 & 150 & 150 & 150 \end{array} $$
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