Question

1. rainwater was collected in water collectors at thirty different sites near an industrial basin and the amount of acidity (ph level) was measured. the mean and standard deviation of the values are 5.2 and 1.8 respectively. when the ph meter was recalibrated back at the laboratory, it was found to be in error. the error can be corrected by adding 0.2 ph units to all of the values and then multiply the result by 1.2. find the mean and standard deviation of the corrected ph measurements.

a) mean = 6.48; standard deviation = 0.36
b) mean = 2.24; standard deviation = 0.36
c) mean = 6.48; standard deviation = 2.16
d) mean = 2.24; standard deviation = 2.16
e) mean = 6.44; standard deviation = 2.36

Explanation:

Step1: Recall the effect of adding a constant

When we add a constant \(c\) to each data - point in a dataset, the mean of the dataset changes by the constant \(c\), but the standard deviation remains the same. Here, \(c = 0.2\). Let the original mean be \(\bar{x}\) and the original standard deviation be \(s\). The original mean \(\bar{x}=5.2\) and the original standard deviation \(s = 1.8\).
The new mean \(\bar{y}=\bar{x}+c\).
\(\bar{y}=5.2 + 0.2=5.4\)
The standard deviation after adding a constant to each data - point does not change, so the new standard deviation \(s_y=s = 1.8\). Then we multiply all values by a factor \(k\).

Step2: Recall the effect of multiplying by a constant

When we multiply each data - point in a dataset by a constant \(k\), the mean of the dataset is multiplied by \(k\) and the standard deviation is also multiplied by \(k\). Let \(k = 1.2\).
The new mean \(\bar{z}=k\bar{y}\).
\(\bar{z}=1.2\times5.4 = 6.48\)
The new standard deviation \(s_z=k\times s_y\).
\(s_z=1.2\times1.8=2.16\)

Answer:

C) mean = 6.48; standard deviation = 2.16