Question

Find the range and use it to approximate the value of s. Then calculate the actual value of \(s\). Is the actual value close to the estimate? The following data give the estimated prices of a 170 -gram can or a 200-gram pouch of water-packed tuna for 14 different brands, based on prices paid nationally in supermarkets. $$ \begin{array}{rrrrrrr} .99 & 1.92 & 1.23 & .85 & .65 & .53 & 1.41 \\ 1.12 & .63 & .67 & .69 & .60 & .60 & .66 \end{array} $$

Short answer

Short Answer: The approximate value of the standard deviation (s) using the range rule of thumb is 0.3475, which is slightly higher than the actual standard deviation of 0.277. This means that the approximation somewhat overestimates the standard deviation in this case.

Step-by-step solution

Find Maximum and Minimum Values

In order to find the maximum and minimum values, we look for the highest and lowest value in the given data. The maximum value is 1.92, and the minimum value is 0.53.

Calculate the Range

The range is the difference between the maximum and minimum values. It can be calculated using the formula, Range = Maximum Value - Minimum Value. Range = 1.92 - 0.53 = 1.39

Approximate s using the Range

The range rule of thumb approximation to standard deviation (s) can be given as follows: s ≈ Range / 4 s ≈ 1.39 / 4 = 0.3475

Calculate the Mean

The mean is the sum of all the values divided by the total number of values. In this case, the total number of values (n) is 14. Mean (µ) = (Σx) / n Mean (µ) = (0.99 + 1.92 + 1.23 + 0.85 + 0.65 + 0.53 + 1.41 + 1.12 + 0.63 + 0.67 + 0.69 + 0.60 + 0.60 + 0.66) / 14 Mean (µ) ≈ 0.919

Calculate the Actual Value of s

The actual standard deviation (s) formula is as follows: s = √((Σ(x-µ)²)/(n-1)) Here, we need to find the sum of squared differences between each value (x) and the mean (µ) and then divide it by (n-1), before finding the square root. ∑(x-µ)² = (0.99-0.919)² + (1.92-0.919)² + ... + (0.66-0.919)² ≈ 1.07004 Now, we can calculate the standard deviation (s). s = √(1.07004 / (14 - 1)) s ≈ 0.277

Compare the Approximate and Actual Value of s

Now, we compare the approximated value of s with the actual value. Approximate s = 0.3475 Actual s = 0.277 The approximate value is close to the actual value, but it slightly overestimates the standard deviation in this case.

Key concepts

Range

In data analysis, the range is an important measure that tells us about the spread of a data set. It is calculated as the difference between the largest and smallest values within the data. To find the range, you simply subtract the minimum value from the maximum value. In the tuna price example, the maximum price is 1.92, while the minimum is 0.53. Thus, the range is calculated as follows:

• Range = Maximum Value - Minimum Value
• Range = 1.92 - 0.53
• Range = 1.39

The range tells us how much the prices vary and helps us understand the distribution and extent of price differences among the brands.

Mean

The mean is a fundamental concept in statistics that represents the average of a data set. It is calculated by adding up all the values, and then dividing by the total number of values. It's a useful measure to find the central tendency of a data set. In our tuna prices, there are 14 different values. To find the mean:

• Add all the prices together: 0.99 + 1.92 + 1.23 + 0.85 + 0.65 + 0.53 + 1.41 + 1.12 + 0.63 + 0.67 + 0.69 + 0.60 + 0.60 + 0.66
• Total sum = 12.866
• Divide by the number of items (14): Mean = 12.866 / 14
• Mean ≈ 0.919

This mean can be used to quickly understand what might be considered a typical or common price among these brands.

Data Analysis

Data analysis involves examining datasets to extract valuable insights and information. It helps us understand patterns and trends. When analyzing data, like our tuna prices, we look at measures such as the range and mean to summarize the data. By understanding these, we can make informed decisions.

In this specific example, calculating both the range and mean paints a basic picture of how tuna prices can vary from brand to brand. Such analysis is essential for comparing results and identifying positive or negative trends. It supports validating a hypothesis, such as determining whether a price lies within a certain range as expected.

Key aspects to consider during data analysis include:

• Variability in data
• Central tendencies (mean, median, mode)
• Identifying outliers

These components are crucial in building a comprehensive understanding of the data.

Approximation

In statistics, approximation is a method used to estimate a value using simple calculations. It often involves using rules of thumb or simpler arithmetic to get an estimated result without going into detailed computations. One common approach is the 'range rule of thumb' for approximating the standard deviation (s).

This rule estimates the standard deviation by dividing the range by 4:

• s ≈ Range / 4
• For our data: s ≈ 1.39 / 4 = 0.3475

While approximations can provide quick insights, it's important to note they aren't perfectly accurate. They serve well for initial estimates, especially when quick decisions are necessary. But for precise results, actual calculations (like finding the exact standard deviation) are needed.

Statistics

Statistics is the branch of mathematics dealing with data collection, analysis, interpretation, and presentation. It provides tools and methodologies for describing, understanding, and drawing conclusions about the data. By using statistical techniques, one can infer important insights from data sets such as those comprising tuna prices.

In our example, statistics helps us determine both approximate and exact measures like the mean and standard deviation, guiding us in interpreting price trends.

Statistical measures frequently used include:

• Mean, Median, Mode (central tendencies)
• Standard Deviation, Variance (measures of dispersion)
• Range (simple measure of spread)

Employing statistical principles allows us to make informed decisions based on data rather than assumptions, enhancing accuracy in fields ranging from business to science.