1. ## Probability/stats problem

Hi,

Can anyone help with the following question please:

A chemist regularly carries out a procedure which involves adding three solvents to a flask. The first, A, is delivered using a pipette in such a way that the volume delivered may be considered to be Normally distributed with mean 100 mL and standard deviation 1 mL. The other two are delivered using two different pipettes; in each case the volume delivered may be considered to be Normally distributed with mean 50 mL and standard deviation 0.5 mL. The three volumes are statistically independent. The total volume, T, is the sum of the three.

(a) Derive the expectation and variance of T, clearly stating any results which are assumed in your deviation.
(b) What is the probability that the total volume will exceed 200.2 mL?
(c) Twenty five such procedures are carried out every day. What is the probability that the deviation from 200 mL of the average of the 25 total volumes exceeds 0.2 mL, in either the positive or negative direction?
(d) What is the probability that less than three of the twenty five totals will exceed 200.2 mL?
(e) A Normal probability plot might be used to verify the assumption of Normality assumed in this question. Explain the rationale underlying such plots.

I have (a) expectation = 100 + 50 + 50 = 200 and variance = 1.5, (b) 1 - .565 = 43.5% but am stuck from (c) onwards.

I'd appreciate any assistance.

Thanks

2. Originally Posted by jackiemoon
Hi,

Can anyone help with the following question please:

A chemist regularly carries out a procedure which involves adding three solvents to a flask. The first, A, is delivered using a pipette in such a way that the volume delivered may be considered to be Normally distributed with mean 100 mL and standard deviation 1 mL. The other two are delivered using two different pipettes; in each case the volume delivered may be considered to be Normally distributed with mean 50 mL and standard deviation 0.5 mL. The three volumes are statistically independent. The total volume, T, is the sum of the three.

(a) Derive the expectation and variance of T, clearly stating any results which are assumed in your deviation.
(b) What is the probability that the total volume will exceed 200.2 mL?
(c) Twenty five such procedures are carried out every day. What is the probability that the deviation from 200 mL of the average of the 25 total volumes exceeds 0.2 mL, in either the positive or negative direction?
(d) What is the probability that less than three of the twenty five totals will exceed 200.2 mL?
(e) A Normal probability plot might be used to verify the assumption of Normality assumed in this question. Explain the rationale underlying such plots.

I have (a) expectation = 100 + 50 + 50 = 200 and variance = 1.5, (b) 1 - .565 = 43.5% but am stuck from (c) onwards.

I'd appreciate any assistance.

Thanks
Apply the following theorem:

If $\displaystyle U$ ~ Normal $\displaystyle (\mu, \sigma)$ and you take a sample of size $\displaystyle n$ then the sample mean has the distribution $\displaystyle \overline{U}$ ~ Normal $\displaystyle \left( \mu, \frac{\sigma}{\sqrt{n}} \right)$.