Statistics: basic descriptive statistics, mean , variance, summation

Statistics Problem Set Aug-21-2012

1. Which of the following formulas measure symmetry of a sample data distribution?

(a)$(1/n) \sum (x-\overline{x})^2$

(b) $ (1/n) \sum (x-\overline{x})^3$

(c)$ (1/n) \sum (x-\overline{x})^4$

(d.) Not listed

2. The following were determined for a sample data: n = 10, min=-2, max= 10, sd = 3,
$\overline{x}=5$. The data is invalid since

$max > \overline{x} + (n-1) sd.$

(b) $min < \overline{x} - (n-1) sd$

(c) median is not specified

(d) None of the above.

3. A sample has current mean mean 4.0 with sample size 10. If 5 is added to all sample values, the recomputed mean will :

stay the same!

(b) will increase by 5.0

(c) will decrease by 5

(d)will increase by 0.5 (e) None of the above

4.A sample has current variance of 2.0 with sample size 10. If 5 is added to all sample values, the recomputed variance will :

stay the same!

(b) will increase by 5.0

(c) will decrease by 5

(d)will increase by 0.5

(d) None of the above.

5. The formula $e^{[(1/n)\sum_{i=1}^{i = n} \ln(x_i)}$ is another way of solving for the

(a) HM

(b) GM

(c) AM

(d) RMS.

6. A new data value 2.5 is added to a sample with mean 3 and sample size 10. The recomputed mean will

increase

(b) decrease

(c) stay the same

(d) None of the above

7. A sample has a sample size 10 and median 50, with all sample values unique. If the minimum value in the sample is removed,
the new median will

stay the same!

(b) increase

(c) decrease

(d) None of the above.

8. Express (-1) + 2 + (-3) + 4 + (-5) + 6 in summation form with index k $ of summation varying from 1 to 6. You cannot use
a variable X as these are not stored in an a vector or array!.

9. If X = c( 4,5,6,7,8), what is the value of $\sum_{i=1}^4 (x_i)(x_{i-1})$?.

10.If X is the same above and Y= c(2,3,1,0,4) what is the value of $\sum_{i=1}^5 {(x_i -\overline{x}) (y_i-\overline{y})}$?


Solutions next week! Enjoy first solving.

Descriptive Statistics

Modified True or False. If the statement is true, write T and if the statement if false , write F and explain why!


Q1. The values computed for a sample, arithmetic mean =-4.5, and root mean square= -4 are valid since the root mean square is always greater than or equal to the arithmetic mean.

Q2. Variances computed from a sample were sample variance = 0.4629, and population variance = 0.5141 are valid.

Q3. The variances computed with a sample size n = 10, population variance = 0.4629, and sample variance = 0.60, are valid.

Q4. Various means computed for a sample: harmonic mean, HM = 0.2, geometric mean, GM = 0.2, arithmetic mean, AM = .3,
root mean square, RMS = .4 are valid since they satisfy \(HM\le GM\le AM\le RMS \).

Q5. If the variance of a sample X is 5.3 then the variance of \(Y = -2 (X + 5)\) is \(2 \cdot 5.3 = 10.6\).

Answers:

1. F. The reasoning may sound valid but the root mean square is never negative!

2. F. Sample variance is always greater than population variance!

3. F. Sample varince may be obtained using the formula: $$\sigma_{n-1}^2= \frac{n}{n-1} \sigma_n^2$$. But $$\frac{10}{9} 0.4629 = 0.5141$$ which is not equal to 0.60.

4. Although the given values does satisfy the inequality for various means, whenever any two of the means are equal, then all means should be equal!

5. If X has the variance \(V_X\), then \(Y = a(X + b)\) will have variance $$a^2 V_X$$. Therefore, the varice of Y will be 4 times the variance of X or 4 (5.3)= 21.2.

Problems with complex numbers.

Q1. Convert the complex number z = 6 + 8j to polar form.

A1. The magnitude \(|z|= \sqrt{6^2 + 8^2}= \sqrt{36+ 64} = \sqrt{100} = 10.\)

Since both x and y components are positive, the complex number is in the first quadrant.
The argument \(\theta = atan(\frac{8}{6}) = 0.9273\), in radians. Multiply by \( 180/\pi \) to obtain the argument degrees. The value would be \(53.3^\circ\).

Q2, Convert to rectangular form the complex number in polar form \( 12 \angle 210^\circ \).

A2. The complex number is in the third quadrant. This means that both components are negative.
\(x = 12 \cos(210^\circ) = -10.392, y = 12 \sin(210^\circ) = -6.0\),from which \(z = -10.392 - 6j.\)

Fomula mass of chemical compound

Q. Calculate the formula mass of the compounds KBr and \( NH_4CN\)

A. The atomic masses of K and Br are respectively 39.102 and 79.904 respectively. Thus the formula mass is 39.102+79.904= 119.006 amu.

On the other hand, NH4CN has a formula mass calculated in the following table:

Component	Amu	Total
N	14.00671	14.0067
H4	1.008(4)	4.032
C	12	12
N	14.0067	14.0067
Total		44.107

The formula mass is then 44.107.

Integrals of the form ∫ xn ex dx by integration by parts (IBP)

These problems originally appeared in my former alternate blog at http://www.optimal-learning-systems.org/eps.


Q1. What is the value of the integral \(\int xe^xdx \)?

A1. Let \(u=x\). Then \(du=dx\). Also let \(dv=e^xdx\). Then \( v= \int e^xdx=e^x \).

Now apply the integration by parts formula, \( \int u dv=uv−\int v du \).

We have, \( \int x e^xdx=x e^x−\int e^xdx=x e^x−e^x=e^x[x−1] \).

Therefore, \( \int xe^xdx=e^x[x−1]\).

One may wonder why the constant of integration is not added in the formula for v. The IBP formula becomes in this case \( u(v+C)−∫(v+C)du\) and one can see that the terms containing the constant of integration cancels out.

Q2. Determine the integral \(\int x^2e^x dx\).

A2. Let \(u=x^2\), then \(du=2^xdx\). Also let \(dv=e^xdx\), then \(v=\int e^xdx\). Then \( dv=e^x dx\). Substituting in the IBP formula,

\( x^2e^x−\int e^x(2x)dx=x^2e^x−2∫xe^xdx\). But the last term was the problem we treated before! Hence, \( \int x^2e^xdx=x2e^x−2[e^x(x−1)]\) and therefore

\[\int x^2e^xdx=e^x [x2−2(x−1)]\]

You may add the constant of integration in the final answer.

@@@@@

Q3. Determine the integral \(\int x^3e^x dx\).

A3. As usual, let \(u=x^3\), then \(du=3x^2dx\). Also let \(dv=e^xdx\), then \(v=e^x\). Substituting in the IBP formula,

\(x^3e^x−\int e^x(3x^2)dx=x^3e^x−3∫x2e^xdx\). But the last term involved an integral we treated before! Therefore \[ \int x^3e^x dx=x^3e^x−3e^x[x^2−2(x−1)]=e^x[x^3−3(x^2−2(x−1))]\]
.
Notice that \[ \int x^ne^x dx=x^ne^x−n\int x^{n−1}e^xdx \]. We leave the fun to the student to discover on his/her own a general formula for the answer.

Calculus: derivative of products of functions of x.

Q. What is the derivative of \( x^2 e^x \sin(x) \) ?


A. The derivative of a product of three functions in \(x\), say, \(u, v \) and \(w\),\( \frac{d(uvw)}{d x} \) is given by

\[ \frac{d(uvw)}{dx} =\frac{uvw}{u} \frac{dw}{dx} + \frac{uvw}{v} \frac{dw}{dv} + \frac{uvw}{u} \frac{dw}{dv}\].

It is easy to generalize this 'product' rule. Now let \( u = x^2, v = e^x, w = \sin{(x)} \). Then we obtain the following result:

\[ e^x \sin(x) \frac{d(x^2)}{dx} + x^2 \sin(x) \frac{d(e^x)}{dx}+ {x^2
e^x}\frac{d(\sin(x) )}{dx} =

2x e^x \sin(x) + x^2 \sin(x) e^x + x^2 e^x \cos(x) \]

Center and radius of a circle from the general equation.

Q. Determine the center and radius of a circle whose equation given by \( x^2 - 4x + y^2 +10y -71 = 0 \).


A. The standard form is given by \( (x-h)^2 + (y-k)^2 = r^2\). Hence we should complete the squares for the terms involving \(x\) and \(y\). For x, the completion of \(x^2 - 4x + \_\_ \) has a constant term given by \( [(-4x/ (2\sqrt{x^2})]^2 = 4\). On the other hand for y, we have \( [10y/(2\sqrt{y^2})]^2= 25\). Thus we obtain \[ (x-\sqrt{4})^2
+(y- -\sqrt{25})^2 -71 - 29 = 0\] which we finalize to \((x -2)^2 + (y-(-5))^2 = 100\). Thus we find the center to be (2, -5) and the radius 10.

Welcome to my new blog. I will post some engineering problems with their solutions here. Stay tuned!