91Ó°ÊÓ

Express all z scores with two decimal places. Plastic Waste Data Set 31 "Garbage Weight" in Appendix B lists weights (ib) of plastic discarded by households. The highest weight is 5.28 Ib, the mean of all of the weights is \(\bar{x}=\) 1.911 Ib, and the standard deviation of the weights is \(s=1.065\) lb. a. What is the difference between the weight of 5.28 Ib and the mean of the weights? b. How many standard deviations is that [the difference found in part (a)]? c. Convert the weight of 5.28 lb to a \(z\) score. d. If we consider weights that convert to \(z\) scores between -2 and 2 to be neither significantly low nor significantly high, is the weight of 5.28 ib significant?

Step by step solution

01

- Find the Difference

To find the difference between the weight of 5.28 lb and the mean, subtract the mean (\bar{x}=1.911) from the given weight. Calculate: \[ \text{Difference} = 5.28 - 1.911 = 3.369 \text{ lb} \]

02

- Find Number of Standard Deviations

To find how many standard deviations the difference is, divide the difference by the standard deviation (\s=1.065). Calculate: \[ \frac{3.369}{1.065} \approx 3.16 \]

03

- Convert to z Score

To convert the weight of 5.28 lb to a \z\ score, use the formula \z = \frac{x - \bar{x}}{s}\, where \x is the value, \ \bar{x}\ is the mean, and \ \ \ is the standard deviation. Calculate: \[ z = \frac{5.28 - 1.911}{1.065} \approx 3.16 \]

04

- Evaluate Significance

Determine if the z score falls within the range of -2 and 2. Since 3.16 is greater than 2, the weight of 5.28 lb is considered significantly high.

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Key Concepts

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

standard deviation

In statistics, understanding the concept of standard deviation is crucial. It's a measure that quantifies the amount of variation or dispersion in a set of values. The standard deviation, denoted as \(s\), tells us how much the individual data points deviate from the mean (average) value.
When we calculate standard deviation, we follow several steps:

• Find the mean (average) of the data set.
• Subtract the mean from each data point and square the result.
• Calculate the average of those squared differences.
• Finally, take the square root of that average.

Standard deviation is pivotal in determining how spread out the data is, which is essential for understanding z scores, variations, and statistical significance. For example, in our exercise, the standard deviation of the weights is 1.065 lb.

mean calculation

The mean, often referred to as the average, is the central value of a data set. It's calculated by summing all the values and then dividing by the number of values.
To find the mean:

• Add up all the numbers in your data set.
• Divide by the number of data points in the set.

As shown in the exercise, the mean weight of the plastic discarded is 1.911 lb.
This calculation is critical since the mean serves as a reference point for other statistical measures, including standard deviation and z scores. A firm grasp of mean calculation helps us interpret various data trends and perform more complex statistical analyses.

statistical significance

Statistical significance helps us understand whether our results are likely to be due to chance or if they indicate a real effect or trend.
In this context, we often use z scores to measure and determine significance:

• If a z score falls within -2 to 2, the data is considered not significantly different from the mean.
• If a z score falls outside this range, it indicates that the data point is significantly different.

For example, the z score of 5.28 lb weight is about 3.16, which is beyond the range of -2 to 2. This tells us that 5.28 lb is a significantly high weight.
Understanding statistical significance is crucial in any analysis as it helps us make informed decisions based on data rather than assumptions.

z score conversion

A z score, or standard score, is a numerical measurement that describes a value's relation to the mean of a group of values. When we convert a value to a z score, we're essentially seeing how many standard deviations the value is from the mean.
The z score formula is \[ z = \frac{x - \bar{x}}{s} \], where:

• \( x \) is the value we're converting
• \( \bar{x} \) is the mean
• \( s \) is the standard deviation

Following this formula, we converted the weight of 5.28 lb to a z score: \[ z = \frac{5.28 - 1.911}{1.065} \approx 3.16 \]
This calculation helps us determine how unusual a particular weight is within the dataset. In our case, a z score of 3.16 indicates the 5.28 lb weight is significantly higher than the average household plastic waste.