Standard Deviation about Mean for Frequency Distributions

Q.

The standard deviation of the numbers $31,32,33........46,47$ is

1. $\sqrt{\frac{17}{12}}$

2. $\sqrt{\frac{({47}^2-1)}{12}}$

3. $2\sqrt{6}$

4. $4\sqrt{3}$

Q.

If the mean and the standard deviation of the data $3,5,7,a,b$ are $5$ and $2$ respectively, then $a$ and $b$ are the roots of the equation

1. $x^2-20x+18=0$

2. $x^2-10x+19=0$

3. $2x^2-20x+19=0$

4. $x^2-10x+18=0$

Q. Four fair dice are thrown independently $27$ times. Then the expected number of times, at least two dice show up a three or five, is

Q.

A random variable $X$ has the probability distribution given below. Find its variance.

$X$	$1$	$2$	$3$	$4$	$5$
$P(X=x)$	$k$	$2k$	$3k$	$2k$	$k$

1. $\frac{16}{3}$

2. $\frac{4}{3}$

3. $\frac{5}{3}$

4. $\frac{10}{3}$

Q.

You want to create a $95\%$ confidence interval with a margin of error of no more than $0.05$ for a population proportion.

The historical data indicates that the population has remained constant at about $0.55$.

What is the minimum size random sample you need to construct this interval?

1. $223$

2. $324$

3. $378$

4. $381$

Q.

Medley in the $4\times 100$ medley relay event, four swimmers swim $100$ yards, each using a different stroke.

A college team preparing for the conference championship looks at the times their swimmers have posted and creates a model based on the following assumptions:
$\bullet$ The swimmers performances are independent.
$\bullet$Each swimmers times follow a Normal model.
$\bullet$The means and standard deviations of the times (in seconds) are as shown:

$$\begin{array}{|ccc|}\hline \text{Swimmer}& \text{Mean}& \text{SD}\\ 1\text{(backstroke)}& 50.72& 0.24\\ 2\text{(breaststroke)}& 55.51& 0.22\\ 3\text{(butterfly)}& 49.43& 0.25\\ 4\text{(freestyle)}& 44.91& 0.21\\ \hline\end{array}$$
a) What are the mean and standard deviation for the relay team’s total time in this event?
b) The team’s best time so far this season was 3:19.48. (That’s 199.48 seconds.)

Do you think the team is likely to swim faster than this at the conference championship? Explain.

Q.

Let$x_1,x_2,x_3,...,x_n$, be $n$ observations and $a$ be the arithmetic mean then the formula for the standard deviation is given by?

Q.

The error in two readings of A are $0.01$ and $-0.03$, then mean absolute error is?

Q. Find the mean, variance and standard deviation for the following frequency distribution.

Classes	0-10	10-20	20-30	30-40	40-50
Frequency	5	8	15	16	6

Q. In a series of $2n$ observations, half of them equal $a$ and remaining half equal $-a$. If the standard deviation of the observations is $2,$ then $|a|$ equals

Q. For the following probability distribution determine standard deviation of the random variable $X$

$X$	2	3	4
$P\left(X\right)$	0.2	0.5	0.3