# Thread: Stats, Standard Deviation

1. ## Stats, Standard Deviation

The average Belfast household uses 14,000 litres of water per week, with a standard deviation of 3,000 litres. Assume a normal distribution.

i. What proportion of Belfast households use less than 14,000 litres of water per week?
ii. What proportion of Belfast households use less than 8,000 litres of water per week?
iii. What proportion of Belfast households use between 8,000 and 12,000 litres of water per week?

2. Originally Posted by tone999
The average Belfast household uses 14,000 litres of water per week, with a standard deviation of 3,000 litres. Assume a normal distribution.

i. What proportion of Belfast households use less than 14,000 litres of water per week?
ii. What proportion of Belfast households use less than 8,000 litres of water per week?
iii. What proportion of Belfast households use between 8,000 and 12,000 litres of water per week?
i. The mean is X = 14,000. Surely you can calculate Pr(X < 14,000) !!

ii. Calculate Pr(X < 8000) = Pr(X > 20,000) by symmetry
= 1 - Pr(X < 20,000).

iii. Calculate Pr(8000 < X < 12,000) = Pr(X < 12,000) - Pr(X < 8000).
Note that Pr(X < 12,000) = Pr(X > 16,000) by symmetry
= 1 - Pr(X < 16,000).

In each case, convert the given X values to Z values and then use your standard normal distribution tables to calculate the required probabilities. You will find many worked examples of this type in this subforum.

If you need more help, please post what you've done and where you get stuck.

3. (i) = 50%

(ii) z = (X - μ) / σ = 8000-14000/3000 = -2

-2 = 0.977 = 1 - 0.977

Ans = 0.023

(iii) z = (X - μ) / σ = 12000-14000/3000 - 0.023
= 0.67 - 0.023
0.67 = 0.749
1-0.749 = 0.251
0.251 - 0.023 = Ans 0.228

Is this correct?

4. Originally Posted by tone999
(i) = 50%

(ii) z = (X - μ) / σ = 8000-14000/3000 = -2

-2 = 0.977 = 1 - 0.977 Mr F says: This makes absolutely no sense.

Ans = 0.023 Mr F says: Correct to 3 decimal places.

(iii) z = (X - μ) / σ = 12000-14000/3000 - 0.023
= 0.67 - 0.023
0.67 = 0.749 Mr F says: This makes absolutely no sense.
1-0.749 = 0.251
0.251 - 0.023 = Ans 0.228 Mr F says: I get 0.2297. I suppose if you're using tables you might get an answer like this. So it's probably right, given where your numbers are coming from, but it's not correct.

Is this correct?
It would help if your working was set out clearly and logically. It is NOT true that -2 = 0.977. And it is NOT true that 0.67 = 0.749 etc. If you handed this exact work in to me I would find it difficult to give it a pass mark.

5. (ii) z = (X - μ) / σ = 8000-14000/3000 = -2

From here do i not just look at my normal distribution table?

2 gives 0.977

(iii) z = (X - μ) / σ = 12000-14000/3000 - 0.023
= 0.67 - 0.023
0.67 = 0.749

Again using the table down to 0.6 and across to 7 gives 0.749 ?

I just read your other information. Sorry it wasnt very tidy, ill improve on that. Could you tell me how to use the table correctly if i am not already doing so?

Is it because I am using a normal distribution table and you are prehaps using a standard normal distrution table?