January 2008 - Question 6



As we are already given the Σx and the n so to get the mean, we just need to divide them.

Σx / n = 180.6 / 12 = 15.05


To work out the standard deviation, we need to work out the Sxx.

Sxx = Σx2 - n (x̄)2


As we have already worked out the Σx, we just need to put that in and square it, then put in the n and times it by the mean we have just worked out squared.

3107.56 - 12(15.05)2 = 389.53


To get the standard deviation, we need to divide the Sxx by n - 1 and then square root it.

s = √Sxx / n - 1
s = √389.53 / 11
s = √35.41181818
s = 5.950782989 = 5.95 (2 d.p.)




To find an outlier, it has to be more than x̄ ± 2s.

s × 2 = 11.9
x̄ + 11.9 = 26.95
x̄ - 11.9 = 3.15


So if the value is below 26.95 or above 3.15, it would be an outlier.
Looking at the table, there is no outliers.




These are 3 easy marks. We just need to get our mean and standard deviation we worked out earlier and put them into the formula.

Mean in degrees Celsius = 15.05
y = 1.8 × 15.05 + 32
y = 59.09

Standard deviation in degrees Celsius = 5.95
y = 1.8 × 5.95 + 32
y = 42.71




This is relatively easy to compare. Just write what you notice.

The New York mean is higher than the Cambridge mean.
The New York temperatures have a greater standard deviation than the Cambridge standard devation.





To help us plot the graph, let's make a table of the upper bound and cumulative frequency...

Upper Bound 70 100 110 120 150 170 190
Cumulative Frequency 0 6 14 24 35 45 48


Now all we have to do is put this into graph form. To get full points your graph should look like this...



Remember to...
• Number your graph suitably, e.g. 70 to 190, 0 to 50 (the 0 to 50 scale should not go over 100).
• Label your axis correctly, e.g. "cumulative frequency" and "hours".
• Plot your points correctly e.g. not the mid-point or lower bound.
• Join the dots together either by line or curve. MUST include (70,0).




To do this, we're going to take 90% of the highest cumulative frequency.

90% of 48 = 43.2


Now we're going to look on our graph where 43.2 touches the line. As you can see it's around 160, our estimate of the 90th percentile is 160.

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