MATHS ASSIGNMENT

Maths assignment

Maths Assignment

Question 1

An experiment carried out twice produces the following results:

12 +\- 2

9 +/- 3:

Calculate the weighted mean result and its uncertainty. What are the 90% confidence limits of the measurement? By x% confidence limits we mean the (usually symmetric) boundaries within which x% of the area falls.

Calculationg weighted mean:

 A weighted average assigns to each measurement xi a weight wi and the average is then

?iwixi?iwi

12+9 = 21

Error uncertainity = 2 + 3 = 5

Total weighted average of measurements = 21 cm

Total error = 5 +/-

Question 2

Joe is making banana cream pie. The recipe calls for exactly 16 ounces of mashed banana. Joe mashes three bananas, and then puts the bowl of pulp onto a scale. After subtracting the weight of the bowl, he finds a value of 15.5 ounces. Not satisified with this answer, he makes several more measurements, removing the bowl from the scale and replacing it between each measurement. Strangely enough, the values he reads from the scale are slightly different each time:

15.5, 16.4, 16.1, 15.9, 16.6 ounces

Joe can calculate the average weight of the bananas:

15.5 + 16.4 + 16.1 + 15.9 + 16.6 ounces

Average = -------------------------------------------

5

= 80.4 ounces / 5 = 16.08 ounces

Now, Joe wants to know just how flaky his scale is. There are two ways he can describe the scatter in his measurements. The mean deviation from the mean is the sum of the absolute values of the differences between each measurement and the average, divided by the number of measurements:

0.5 + 0.4 + 0.1 + 0.1 + 0.6 ounces

Mean dev from mean = --------------------------------------

5

= 1.6 ounces / 5 = 0.32 ounces

The standard deviation from the mean is the square root of the sum of the squares of the differences between each measurement and the average, divided by one less than the number of measurements:

[(0.5) ^2 + (0.4) ^2 + (0.1) ^2 + (0.1) ^2 + 0.6) ^2]

Stdev from mean = sqrt [-----------------------------------------------]

5 - 1

[0.79 ounces^2]

= sqrt [--------------]

[4]

= 0.44 ounces

Either the mean deviation from the mean, or the standard deviation from the mean, gives a reasonable description of the scatter of data around its mean value.

Can Joe use his mashed banana to make the pie? Well, based on his measurements, he estimates that the true weight of his bowlful is (using mean deviation from the mean)

16.08 - 0.32 ounces < true weight < 16.08 + ...