Question 1162753
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Problem 1


Here is the original table
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td></tr><tr><td>80 <= c < 100</td><td>4</td></tr><tr><td>100 <= c < 120</td><td>7</td></tr><tr><td>120 <= c < 140</td><td>13</td></tr><tr><td>140 <= c < 160</td><td>16</td></tr><tr><td>160 <= c < 180</td><td>7</td></tr><tr><td>180 <= c < 200</td><td>3</td></tr></table>
where c = cost of electricity and f represents the frequency that goes with it. When I write something like 80 <= c < 100, I mean that c is between 80 and 100. Furthermore, c could be 80, but c cannot be 100.


Form a new column of values that are the midpoints of the intervals given. The midpoint between 80 and 100 is 90, since (80+100)/2 = 180/2 = 90. The midpoint of 100 and 120 is 110 for similar reasoning. You could repeatedly apply the midpoint formula, or note that the midpoints 90, 110, ... are going up by 20. This is because the start points and endpoints of each interval also increase by 20. Everything is in lockstep together.


Here's what the table should look like after you have added the midpoint column. I'm making x be the midpoint values
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td><td>midpoint cost (x)</td></tr><tr><td>80 <= c < 100</td><td>4</td><td>90</td></tr><tr><td>100 <= c < 120</td><td>7</td><td>110</td></tr><tr><td>120 <= c < 140</td><td>13</td><td>130</td></tr><tr><td>140 <= c < 160</td><td>16</td><td>150</td></tr><tr><td>160 <= c < 180</td><td>7</td><td>170</td></tr><tr><td>180 <= c < 200</td><td>3</td><td>190</td></tr></table>
The midpoint is a representative of the entire interval. For some interval like 80 <= c < 100, we don't know what the exact value of c is (as c is likely to vary anyway). So the next best thing is to just pick the most representative value from the interval. The center is the best bet.


Next step is to multiply the corresponding x and f values to form a new column x*f
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td><td>midpoint cost (x)</td><td>x*f</td></tr><tr><td>80 <= c < 100</td><td>4</td><td>90</td><td>360</td></tr><tr><td>100 <= c < 120</td><td>7</td><td>110</td><td>770</td></tr><tr><td>120 <= c < 140</td><td>13</td><td>130</td><td>1690</td></tr><tr><td>140 <= c < 160</td><td>16</td><td>150</td><td>2400</td></tr><tr><td>160 <= c < 180</td><td>7</td><td>170</td><td>1190</td></tr><tr><td>180 <= c < 200</td><td>3</td><td>190</td><td>570</td></tr></table>
For example, in row 2 we have x*f = 110*7 = 770. The other rows are treated the same.


We're almost done. Add up everything in the x*f column
360+770+1690+2400+1190+570 = 6980


Finally, divide that sum by n = 50 because the sum of the frequency (f) values is 50. This will yield the mean
mean = (sum of x*f)/(sum of f)
mean = 6980/50
mean = 139.6


We will use the mean later for problem 2. We'll call the mean m, so m = 139.6


Answer: estimated mean cost is approximately 139.6 shekels


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Problem 2


Go back to the original table
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td></tr><tr><td>80 <= c < 100</td><td>4</td></tr><tr><td>100 <= c < 120</td><td>7</td></tr><tr><td>120 <= c < 140</td><td>13</td></tr><tr><td>140 <= c < 160</td><td>16</td></tr><tr><td>160 <= c < 180</td><td>7</td></tr><tr><td>180 <= c < 200</td><td>3</td></tr></table>


Subtract the mean (m) from each x value. Then square the result. Lastly, multiply that square by the frequency f value
This will form a new column labeled f*(x-m)^2
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td><td>midpoint cost (x)</td><td>f*(x-m)^2</td></tr><tr><td>80 <= c < 100</td><td>4</td><td>90</td><td>9840.64</td></tr><tr><td>100 <= c < 120</td><td>7</td><td>110</td><td>6133.12</td></tr><tr><td>120 <= c < 140</td><td>13</td><td>130</td><td>1198.08</td></tr><tr><td>140 <= c < 160</td><td>16</td><td>150</td><td>1730.56</td></tr><tr><td>160 <= c < 180</td><td>7</td><td>170</td><td>6469.12</td></tr><tr><td>180 <= c < 200</td><td>3</td><td>190</td><td>7620.48</td></tr></table>

For instance, row 3 has the following calculation
f*(x-m)^2 = 13*(130-139.6)^2 = 1198.08


Add up everything in the f*(x-m)^2 column to get
9840.64+6133.12+1198.08+1730.56+6469.12+7620.48 = 32,992 


Divide this over n-1 = 50-1 = 49 to get the sample variance
sample variance = 32992/49 = 673.30612244898


Apply the square root to get the sample standard deviation
sqrt( 673.30612244898 ) = 25.9481429479833



Answer: Approximately 25.9481429479833
Round this value however you need to


Side note: we want the sample standard deviation (rather than the population standard deviation) because our sample of n = 50 families represents a larger population in which we don't know about. We only use a population standard deviation if we have data on the entire population (ie a census).


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Problem 3


Original table
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td></tr><tr><td>80 <= c < 100</td><td>4</td></tr><tr><td>100 <= c < 120</td><td>7</td></tr><tr><td>120 <= c < 140</td><td>13</td></tr><tr><td>140 <= c < 160</td><td>16</td></tr><tr><td>160 <= c < 180</td><td>7</td></tr><tr><td>180 <= c < 200</td><td>3</td></tr></table>



Highlight the rows where the cost c is between 140 and 200
<table border = "1" cellpadding = "5"><tr><td>Electricity Cost (c)</td><td>Frequency (f)</td></tr><tr><td>80 <= c < 100</td><td>4</td></tr><tr><td>100 <= c < 120</td><td>7</td></tr><tr><td>120 <= c < 140</td><td>13</td></tr><tr><td><font color=red>140 <= c < 160</font></td><td><font color=red>16</font></td></tr><tr><td><font color=red>160 <= c < 180</font></td><td><font color=red>7</font></td></tr><tr><td><font color=red>180 <= c < 200</font></td><td><font color=red>3</font></td></tr></table>
The frequencies here are 16, 7 and 3 which add to 16+7+3 = 26


We have 26 families that fit in this range, out of 50 families total. So approximately 26/50 = 52/100 = 0.52 = 52% of the families pay in the range of 140 to 180 shekels. This is of course an estimate because a sample is an estimate or approximation of the population.


Answer: About 52%
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