91Ó°ÊÓ

The acidity or alkalinity of a solution is measured using \(\mathrm{pH}\). A pH less than 7 is acidic, while a pH greater than 7 is alkaline. The following data represent the \(\mathrm{pH}\) in samples of bottled water and tap water. $$ \begin{array}{lllllll} \hline \text { Tap } & 7.64 & 7.45 & 7.47 & 7.50 & 7.68 & 7.69 \\ & 7.45 & 7.10 & 7.56 & 7.47 & 7.52 & 7.47 \\ \hline \text { Bottled } & 5.15 & 5.09 & 5.26 & 5.20 & 5.02 & 5.23 \\ & 5.28 & 5.26 & 5.13 & 5.26 & 5.21 & 5.24 \\ \hline \end{array} $$ (a) Which type of water has more dispersion in pH using the range as the measure of dispersion? (b) Which type of water has more dispersion in pH using the standard deviation as the measure of dispersion?

Step by step solution

01

- Identify the pH values

List all the pH values for both tap water and bottled water.For tap water: 7.64, 7.45, 7.47, 7.50, 7.68, 7.69, 7.45, 7.10, 7.56, 7.47, 7.52, 7.47.For bottled water: 5.15, 5.09, 5.26, 5.20, 5.02, 5.23, 5.28, 5.26, 5.13, 5.26, 5.21, 5.24.

02

- Calculate the range for tap water

Find the minimum and maximum pH values for tap water.Minimum pH for tap water: 7.10Maximum pH for tap water: 7.69Range = Maximum - Minimum = 7.69 - 7.10 = 0.59

03

- Calculate the range for bottled water

Find the minimum and maximum pH values for bottled water.Minimum pH for bottled water: 5.02Maximum pH for bottled water: 5.28Range = Maximum - Minimum = 5.28 - 5.02 = 0.26

04

- Compare the ranges

By comparing the ranges calculated, we see that tap water has a range of 0.59, while bottled water has a range of 0.26.Therefore, tap water has more dispersion using the range as the measure of dispersion.

05

- Calculate the standard deviation for tap water

Use the formula for standard deviation and calculate it for the tap water values.Step-by-step calculation:1. Calculate the mean: \( \text{Mean}_{\text{tap}} = \frac{7.64 + 7.45 + 7.47 + 7.50 + 7.68 + 7.69 + 7.45 + 7.10 + 7.56 + 7.47 + 7.52 + 7.47}{12} \approx 7.5183 \)2. Calculate each squared difference from the mean.3. Find the average of these squared differences.4. Take the square root of that average.Standard deviation for tap water approximately equals 0.155.

06

- Calculate the standard deviation for bottled water

Use the formula for standard deviation and calculate it for the bottled water values.Step-by-step calculation:1. Calculate the mean: \( \text{Mean}_{\text{bottled}} = \frac{5.15 + 5.09 + 5.26 + 5.20 + 5.02 + 5.23 + 5.28 + 5.26 + 5.13 + 5.26 + 5.21 + 5.24}{12} \approx 5.1933 \)2. Calculate each squared difference from the mean.3. Find the average of these squared differences.4. Take the square root of that average.Standard deviation for bottled water approximately equals 0.090.

07

- Compare the standard deviations

By comparing the standard deviations calculated, we find that the standard deviation for tap water is approximately 0.155, while for bottled water it is approximately 0.090.Therefore, tap water has more dispersion using the standard deviation as the measure of dispersion.

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Key Concepts

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

Range as Measure of Dispersion

The range is one of the simplest ways to measure dispersion in a data set. It helps us understand how spread out the values are. To calculate the range, you subtract the smallest value from the largest value in the data set.

For tap water, the minimum pH value is 7.10 and the maximum pH value is 7.69. So, the range is 7.69 - 7.10 = 0.59. For bottled water, the minimum pH value is 5.02 and the maximum pH value is 5.28. Therefore, the range is 5.28 - 5.02 = 0.26.

Comparing these ranges, we see that tap water has a range of 0.59, while bottled water has a smaller range of 0.26. This means there is more variability in pH levels in tap water. Using the range as the measure, we conclude that tap water has more dispersion.

Standard Deviation Calculation

Standard deviation is a more precise way to measure dispersion, as it considers each data point's deviation from the mean.

To calculate the standard deviation, follow these steps:

• Calculate the mean (average) of the data set. For tap water, the mean pH is approximately 7.5183. For bottled water, it is approximately 5.1933.
• Subtract the mean from each value to find the deviation for each data point.
• Square each deviation to avoid negative values.
• Find the average of these squared deviations.
• Take the square root of that average, which gives the standard deviation.

For tap water, this process gives a standard deviation of approximately 0.155. For bottled water, the standard deviation is about 0.090.

Therefore, the larger standard deviation for tap water indicates that its pH values are more spread out compared to bottled water.

Data Comparison

When we compare data sets, it's essential to use appropriate measures of dispersion to understand their variability. The range provides a quick but less detailed measure, while the standard deviation offers a deeper insight into data spread.

In our pH analysis:

• Tap water has a range of 0.59 and a standard deviation of 0.155.
• Bottled water has a range of 0.26 and a standard deviation of 0.090.

Both the range and standard deviation indicate that tap water has greater dispersion compared to bottled water. This implies tap water pH values are more varied, making its pH levels less consistent than those of bottled water.

Understanding these measures helps us evaluate the reliability and quality of water samples, guiding us in making informed decisions about water consumption.