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Chapter 3: Problem 11

Consider a value to be significantly low if its \(z\) 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
Values 鈮� 5.51542 grams are significantly low; values 鈮� 5.76318 grams are significantly high.

Step by step solution

01

Understand the criteria for significance

A value is considered significantly low if its z-score is less than or equal to -2 and significantly high if its z-score is greater than or equal to 2.
02

Recall the z-score formula

The z-score is calculated using the formula: \[ z = \frac{x - \mu}{\sigma} \]where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
03

Calculate z-scores for significantly low values

For a weight to be significantly low, \( z \leq -2 \). Using the z-score formula: \[ -2 = \frac{x - 5.63930}{0.06194} \]Solving for \( x \): \[ x = -2 \times 0.06194 + 5.63930 \]\[ x \approx 5.51542 \]So, any value less than or equal to 5.51542 grams is considered significantly low.
04

Calculate z-scores for significantly high values

For a weight to be significantly high, \( z \geq 2 \). Using the z-score formula: \[ 2 = \frac{x - 5.63930}{0.06194} \]Solving for \( x \): \[ x = 2 \times 0.06194 + 5.63930 \]\[ x \approx 5.76318 \]So, any value greater than or equal to 5.76318 grams is considered significantly high.
05

Conclusion

Values less than or equal to 5.51542 grams are significantly low, while values greater than or equal to 5.76318 grams are significantly high.

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Key Concepts

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

Significance Levels
In statistics, significance levels help us determine whether a value is considered extreme compared to the rest of the data set. By setting clear thresholds, we can identify values that stand out in a meaningful way.
In the context of the provided exercise, we use z-scores to decide if a coin鈥檚 weight is significantly low or high.
A z-score standardizes the value by showing how many standard deviations it is from the mean \(\text{mean}\). Here, a z-score of \(\text{2 or -2}\) is used as the benchmark.
- If a z-score is less than or equal to \(-2\), the value is significantly low.
- If a z-score is greater than or equal to 2, the value is significantly high.
Understanding these criteria helps in identifying and analyzing outliers in the dataset. Think of significance levels as markers that tell us, 'Hey, this value is way different from what we usually see!'
Mean and Standard Deviation
The mean and standard deviation are fundamental concepts in statistics. They help to summarize and understand data sets.
- The **mean** (\text{also called the average}) is the central value of a dataset. It provides a quick snapshot of the dataset's central tendency. For example, the mean weight of quarters in our dataset is \(5.63930 \text{g}\).
- The **standard deviation** gives us an idea of the spread or variability in the dataset. It tells us how much the values typically deviate from the mean. For the quarters, the standard deviation is \(0.06194 \text{g}\).
Using these two measures, we can compute the z-scores to see how far individual data points lie from the mean and determine if they are extreme.
Data Set Analysis
Analyzing a dataset involves examining its values to draw relevant conclusions. By calculating z-scores for each value in the dataset, we can see how each value compares to the mean and identify outliers.
Steps to analyze the dataset:
1. **Calculate the mean** (\text{center of the data}).
2. **Determine the standard deviation** (\text{spread of the data}).
3. **Compute z-scores** to standardize values.
For this exercise, a z-score reveals how far a coin's weight is from the mean in terms of standard deviations.
- **Low z-scores** (\(\text{-2 or lower}\)) highlight coins significantly lighter than average.
- **High z-scores** (\(\text{2 or higher}\)) point to coins significantly heavier than average.
This process helps in identifying extreme values which could be due to errors, anomalies, or important deviations worth investigating further.

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Most popular questions from this chapter

A student of the author earned grades of A, C. B, A, and 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: \(A=4 ; B=3 ; C=2 ; D=1 ; F=0 .\) 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?

Express all z scores with two decimal places. Pulse rates of adult females are listed in Data Set 1 "Body Data" in Appendix B. The lowest pulse rate is 36 beats per minute, the mean of the listed pulse rates is \(\bar{x}=74.0\) beats per minute, and their standard deviation is \(s=12.5\) beats per minute. a. What is the difference between the pulse rate of 36 beats per minute and the mean pulse rate of the females? b. How many standard deviations is that [the difference found in part (a)]? c. Convert the pulse rate of 36 beats per minutes to a \(z\) score. d. If we consider pulse rates that convert to z scores between -2 and 2 to be neither significantly low nor significantly high, is the pulse rate of 36 beats per minute significant?

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-I, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are the numbers of heroic firefighters who lost their lives in the United States each year while fighting forest fires. The numbers 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 variation? $$\begin{array}{cccccccccccccc} 20 & 18 & 23 & 30 & 20 & 12 & 24 & 9 & 25 & 15 & 8 & 11 & 15 & 34 \end{array}$$

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 annual U.S. sales of vinyl record albums (millions of units). The numbers of albums sold are listed in chronological order, and the last entry represents the most recent year. Do the measures of center give us any information about a changing trend over time? \(\begin{array}{rrrrrrr}0.3 & 0.6 & 0.8 & 1.1 & 1.1 & 1.4 & 1.4 & 1.5 & 1.2 & 1.3 & 1.4 & 1.2 & 0.9 & 0.9\end{array}$$\begin{array}{rrrrrr}1 & 1.9 & 2.5 & 2.8 & 3.9 & 4.6 & 6.1\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 weights in pounds of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII (the same players from the preceding exercise). Are the results likely to be representative of all National Football League (NFL) players? $$\begin{array}{rrrrrrrrr} 189 & 254 & 235 & 225 & 190 & 305 & 195 & 202 & 190 & 252 & 305\end{array}$$
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