TF ErrAna Homework

Errors

Find 3 published examples of data which is wrongly represented.

Identify what is incorrect about it. What does it mean to be wrongly presented? A typical example is a political poll which does not identify the statistical uncertainty properly or at all.

Create a Histogram using ROOT

some commands that may interest you

root [1] TH1F *Hist1=new TH1F("Hist1","Hist1",50,-0.5,49.5);
root [2] Hist1->Fill(10);
root [3] Hist1->Draw();

You can use the above commands but you need to change the names and numbers above to receive credit. You must also add a title to the histogram which contains your full name. You will printout the histogram and hand it in with the above two problems.

Notice how the square rectangle in the histogram is centered at 10!

Notice that if you do the commands

root [2] Hist1->Fill(10);
root [3] Hist1->Draw();

the rectangle centered a 10 will reach the value of 2 on the vertical axis.

Two dice are rolled 20 times. Create a histogram to represent the 20 trials below

Trial	Value
1	8
2	10
3	9
4	5
5	9
6	6
7	5
8	6
9	3
10	9
11	8
12	5
13	8
14	10
15	8
16	11
17	12
18	6
19	7
20	8

Mean and SD

Electron radius

The probability that an electron is a distance [math]r[/math] from the center of the hydrogen atom

[math]P(r) = Cr^2 \exp^{-2 \frac{r}{R}}[/math]

Doing the integrals by hand (no tables) ,

a.)Find the value of C

b.) Find the mean electron radius and standard deviation for an electron in a hydrogen atom

Histograms by Hand

Given the following test scores from 40 students.

Trial	Value	Trial	Value	Trial	Value	Trial	Value
1	49	11	90	21	69	31	74
2	80	12	84	22	69	32	86
3	84	13	59	23	53	33	78
4	73	14	56	24	55	34	55
5	89	15	62	25	77	35	66
6	78	16	53	26	82	36	60
7	78	17	83	27	81	37	68
8	92	18	81	28	76	38	92
9	56	19	65	29	79	39	87
10	85	20	81	30	83	40	86

a.) calculate the mean and standard deviation

b.) construct a histogram by hand which has 10 bins centered on 10,20,...

c.) Use ROOT to construct a histogram. Compare the mean and RMS from ROOT with your result in part (a) above. What is the difference between the RMS report in the ROOT histogram and the standard deviation you calculated in part (a)?

Variance using Probability function

Given that

[math]\sigma^2 = \lim_{N \rightarrow \infty} \frac{1}{N}\sum_{j=1}^n \left [ \left (x_j - \mu \right)^2) \right ][/math]

Justify that

[math]\lim_{N \rightarrow \infty} \frac{1}{N}\sum_{j=1}^n \left [ \left (x_j - \mu \right)^2 \right ] = \lim_{N \rightarrow \infty} \left [ \frac{1}{N}\sum_{j=1}^n x_j^2\right ] - \mu^2[/math]

Note

The standard deviation ([math]\sigma[/math]) is the root mean square (RMS) of the deviations.

RMS = [math]\sqrt{\frac{1}{N}\sum_i^N x_i^2}[/math] so [math]\sigma = \mbox{RMS}(x_i -\mu)[/math]

Binomial Probability Distributions

1.)Evaluate the following (at least one by hand)

a.) [math]{6\choose 3}[/math]

b.) [math]{4\choose 2}[/math]

c.) [math]{10\choose 3}[/math]

d.) [math]{52\choose 4}[/math]

2.) Plot the binomial distribution P(x) for n=6 and p=1/2 from x=0 to 6.

[math]P(x) = {n\choose x}p^{x}q^{n-x} [/math]

3.) Given the probability distribution below for the sum of the point on a pair of dice

[math]P(x) =\left \{ {\frac{x-1}{36} \;\;\;\; 2 \le x \le 7 \atop \frac{13-x}{36} \;\;\; 7 \le x \le 12} \right .[/math]

a.) find the mean

b.) find the standard deviation [math](\sigma)[/math]

4.) Prove that [math]\sigma^2 = npq[/math] for the Binomial distribtuion.

Error Propagation

[1] Forest_Error_Analysis_for_the_Physical_Sciences