Water hardness is a traditional measure of the capacity of water to react with s
ID: 3260687 • Letter: W
Question
Water hardness is a traditional measure of the capacity of water to react with soap. Hardness in water is caused by dissolved calcium and magnesium. It is expressed as the equivalent quantity of calcium carbonate. Fifteen measurements were collected from randomly chosen lakes in a particular district. The mean hardness is 102.03 mg/l and the estimated standard error of the mean is 1.378 mg/l.
a) Give the smaple standard deviation of the 15 hardness measurements.
b) Assuming that water hardness is normally distributed, compare a 95% confidence interval for the mean water hardness.
c) According to Health Canada, water with a hardness smaller than 100 mg/l is classified as soft and water with a hardness larger than 180 mg/l is classified as hard. If the hardness is between soft and hard, we say that the water is medium hard. With a level of confidence of 95%, can we classify the mean hardness of the water in this district as medium hard?
d) Question c) but use a level of confidence of 80% instead.
Thank you in advance!
Explanation / Answer
M = 102.03
Standard error = 1.378
.
Answer to part a)
Sample standard deviation = Standard error * square root (sample size)
Sample standard deviation = 1.378 * sqrt(15)
Sample standard deviation = 5.3370
.
Answer to part b)
The formula of 95% confidence interval is as follows:
M - Z*SE , M + Z * SE
.
value of Z for 95% confidence interval is : 1.96
.
On plugging the values we get:
102.03 -1.96 * 1.378 , 102.03 +1.96 *1.378
99.3291 , 104.73088
.
Answer to part c)
No , based on the working on part b we can nott clasify the 95% confidence interval of 99.33 -104.73 as medium hard water , because the lower limit of this interval falls in the range of soft water.
.
Answer to part d)
for 80% , the value of Z is :1.28
.
On plugging it in the formula we get
102.03 - 1.28 * 1.378 , 102.03 + 1.28 * 1.378
100.2662 , 103.7938
Thus , yes we can now classify the 80% interval int he medium hard range of water
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