Question

Chapter 3 – Numerically Summarizing Data 1. Section 3.1 Measures of Central Tendency pH in Water: The acidity of alkalinity of a solution is measured using pH. A pH less than 7 is acidic; a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. 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 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 a) Determine the mean, median and mode pH for each type of water. b) Comment on the differences between the two water types. c) Suppose the pH of 7.10 in tap water was incorrectly recorded as 1.70. How does this affect the mean

Answer 1

Answer with Step-by-step explanation:

For the tap water the observations are summarized as below:

Tap water sample:

$\bar{pH}_{tap}=\frac{\sum_{i=1}^{N}x_i}{N}$

where

$x_i$ is the $i^{th}$ observation

'N' is the total number of observations

Applying the given values we get

$\bar{pH}=\frac{7.64+7.45+7.47+7.50+7.68+7.69+7.45+7.10+7.56+7.47+7.52+7.47}{12}\\\\\bar{pH}_{tap}=7.5$

To obtain the median of the given data arranging the given data in ascending order we get

7.10, 7.45,7.45, 7.47,7.47,7.47,7.50, 7.52, 7.56,7.64, 7.68,7.69

since the data is even thus the median of the given data is $Median=\frac{7.47+7.50}{2}=7.485$

Mode of the data is the value that occurs most frequently thus for the given data mode is 7.47

Part b)

Bottled water sample:

$\bar{pH}_{bottled}=\frac{5.15+ 5.09+ 5.26+ 5.20+ 5.02+ 5.23+ 5.28+ 5.26+ 5.13+ 5.26+ 5.21+ 5.24}{12}\\\\\bar{pH}_{bottle}=5.194$

To obtain the median of the given data arranging the given data in ascending order we get

5.02, 5.09,5.13, 5.15,5.20,5.21,5.23, 5.24, 5.26,5.26, 5.26,5.28

since the data is even thus the median of the given data is $Median=\frac{5.21+5.23}{2}=5.422$

Mode of the data is the value that occurs most frequently thus for the given data mode is 5.26

Part c)

If the 7.10 water measurement is taken incorrectly as 1.70 the mean changes to

$\bar{pH}=\frac{7.64+7.45+7.47+7.50+7.68+7.69+7.45+1.70+7.56+7.47+7.52+7.47}{12}\\\\\bar{pH}_{tap}=7.05$