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Math  /  Data & Statistics

QuestionQhenstion 1 [16] A sample at 15 hothehotds in T thwane wore surveyod to identify their average water usage per month (it) Kiloitres). The usage per househods were given below \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline 24 & 32 & 14 & 26 & 15 & 10 & 37 & 16 & 14 & 17 & 19 & 24 & 20 & 19 & 33 \\ \hline \end{tabular} 1 1 Calculiate the sample mean water usage for the sample I E Catculate the median water usage for the sample (2) 1 i f find the first and the second quartite for the sample. (3) 1.3 Ciaturate the sample standard deviation for the is househots. (6) (S)(S)

Studdy Solution

STEP 1

1. The sample consists of 15 households.
2. The water usage data for the households are: 24, 32, 14, 26, 15, 10, 37, 16, 14, 17, 19, 24, 20, 19, 33.
3. The sample mean, median, quartiles, and standard deviation need to be calculated.

STEP 2

1. Calculate the sample mean water usage.
2. Calculate the median water usage.
3. Determine the first (Q1) and second (Q2) quartiles.
4. Calculate the sample standard deviation.

STEP 3

First, sum all the water usage values: 24+32+14+26+15+10+37+16+14+17+19+24+20+19+33=320 24 + 32 + 14 + 26 + 15 + 10 + 37 + 16 + 14 + 17 + 19 + 24 + 20 + 19 + 33 = 320

STEP 4

Now, divide the sum by the number of households (15) to find the sample mean: Sample Mean=32015=21.33 kilolitres \text{Sample Mean} = \frac{320}{15} = 21.33 \, \text{kilolitres}

STEP 5

Sort the data in ascending order: 10,14,14,15,16,17,19,19,20,24,24,26,32,33,37 10, 14, 14, 15, 16, 17, 19, 19, 20, 24, 24, 26, 32, 33, 37

STEP 6

Identify the median, which is the middle value of the sorted data. Since there are 15 values, the median is the 8th value: Median=19 kilolitres \text{Median} = 19 \, \text{kilolitres}

STEP 7

Find the first quartile (Q1), which is the median of the first half of the data (not including the overall median if the number of data points is odd): The first half of the sorted data is: 10,14,14,15,16,17,19 10, 14, 14, 15, 16, 17, 19 The median of this subset is the 4th value: Q1=15 kilolitres Q1 = 15 \, \text{kilolitres}

STEP 8

Find the third quartile (Q3), which is the median of the second half of the data (not including the overall median if the number of data points is odd): The second half of the sorted data is: 19,20,24,24,26,32,33,37 19, 20, 24, 24, 26, 32, 33, 37 The median of this subset is the 4th value: Q3=26 kilolitres Q3 = 26 \, \text{kilolitres}

STEP 9

Calculate the sample standard deviation using the formula: s=∑i=1n(xi−xˉ)2n−1 s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} where xˉ\bar{x} is the sample mean and nn is the number of observations.
First, compute each (xi−xˉ)2(x_i - \bar{x})^2: (24−21.33)2=7.11,(32−21.33)2=113.56,(14−21.33)2=53.78, (24 - 21.33)^2 = 7.11, \quad (32 - 21.33)^2 = 113.56, \quad (14 - 21.33)^2 = 53.78, (26−21.33)2=21.89,(15−21.33)2=40.11,(10−21.33)2=128.89, (26 - 21.33)^2 = 21.89, \quad (15 - 21.33)^2 = 40.11, \quad (10 - 21.33)^2 = 128.89, (37−21.33)2=245.78,(16−21.33)2=28.44,(14−21.33)2=53.78, (37 - 21.33)^2 = 245.78, \quad (16 - 21.33)^2 = 28.44, \quad (14 - 21.33)^2 = 53.78, (17−21.33)2=18.78,(19−21.33)2=5.44,(24−21.33)2=7.11, (17 - 21.33)^2 = 18.78, \quad (19 - 21.33)^2 = 5.44, \quad (24 - 21.33)^2 = 7.11, (20−21.33)2=1.78,(19−21.33)2=5.44,(33−21.33)2=136.89 (20 - 21.33)^2 = 1.78, \quad (19 - 21.33)^2 = 5.44, \quad (33 - 21.33)^2 = 136.89

STEP 10

Sum all the squared deviations: 7.11+113.56+53.78+21.89+40.11+128.89+245.78+28.44+53.78+18.78+5.44+7.11+1.78+5.44+136.89=868.78 7.11 + 113.56 + 53.78 + 21.89 + 40.11 + 128.89 + 245.78 + 28.44 + 53.78 + 18.78 + 5.44 + 7.11 + 1.78 + 5.44 + 136.89 = 868.78

STEP 11

Divide the sum by n−1n-1 (which is 14): 868.7814=62.06 \frac{868.78}{14} = 62.06

STEP 12

Take the square root of the result to find the sample standard deviation: s=62.06≈7.88 kilolitres s = \sqrt{62.06} \approx 7.88 \, \text{kilolitres}
Solution:
- Sample Mean: 21.3321.33 kilolitres - Median: 1919 kilolitres - First Quartile (Q1): 1515 kilolitres - Third Quartile (Q3): 2626 kilolitres - Sample Standard Deviation: 7.887.88 kilolitres

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