Download 1) How many significant digits are there in the following numbers? a

1) How many significant digits are there in the following numbers?
a) 976.45 b)84,000 c) 0.0094 d)301.07 e) 4.000 f) 5280
5
2
2
5
4
3
2) What is the most significant digit in each of the numbers of problem 1.
a) 9 b) 8 c)9 d) 3 e)4 f)5
3) Round each number in problem 1 to two significant digits.
a)980 b) 84000 c) 0.0094 d) 3.0 x102 e) 4.0 f) 5300
4) The results of 20 rolls of a pair of dice are shown below.
Calculate the mean, mode, and median from this data.
1. 3
2. 7
3. 4
4. 8
5. 12
6. 8
7. 9
8. 7
9. 5
10. 7
number rolled
1
2
3
4
5
6
7
8
9
10
11
12
11. 12
12. 8
13. 6
14. 6
15. 7
16.
17.
18.
19.
20.
6
7
8
9
8
number of times rolled
0
0
1
1
1
3
5
5
2
0
0
2
median: 7
mode: The mode is not unique: 7 and 8 are both at the highest frequency.
mean: 7.35
5) A set of grades are reported from an exam.
What is average and standard deviation of these data.
Grades: 80, 81, 67, 90, 91, 68, 99, 87, 77, 76, 89, 100, 100, 75, 74
mean = 83.6
stdDev = 10.7 (if I used N in the formula (N=15))
stdDev = 11.1 (if I use N-1 in the formula (N-1=14))
The second calculation is the technically correct answer since
I used the data to calculate mean. The difference between the
even with N=15 is fairly small (few %).
6) Calculate the weighted mean for the following data:
x1 = 3.01 + .10
x2 = 3.03 + .24
x3 = 3.00 + .05
mean = 3.00
7) The formula for the weighted mean is:
x=
∑w x
∑w
i
i
i
Where wi =
1
σ i2
a)Using the propagation of error formula derive the uncertainty on the mean.
See lecture 4
1
σx =
∑ wi
b)Assuming all σi are equal show that this formula reduces to the error on the
unweighted average that was derived in class.
σx =
1
=
∑ wi
∑
1
1
σ i2
=
∑
1
1
σ2
=σ
1
∑1
=σ
1
σ
=
N
N
8) A quantity x is measured with an uncertainty of σx.
A quantity y is derived from x using the formulae below.
Quote the error on y in terms of σx for each equation (propagate the errors).
a) y=ax+b
σ y = aσ x
b) y= 1/x
σy =
σx
x2
2
c) y= a*x
σ y = 2axσ x
d) y= ln(x/d)
σy =
σx
x
e) y = a*exp(-x/d)
a
x
σ y = exp(− )σ x
d
d