Descriptive Statistics Test 2

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Descriptive Statistics Test 2

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Question 1
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The variance of first $$ 50$$ even natural numbers is

A
$$ \displaystyle \frac{833}{4}$$

B
$$ 833$$

C
$$ 437$$

D
$$ \displaystyle \frac{437}{4}$$

Solution

$${\textbf{Step-1: Use mean formula and find the mean.}}$$
$$\Rightarrow \sigma^2 = \dfrac{1}{n} \sum x_i^2 - \bar{(x)}^2$$

$$\Rightarrow n = 50, \sum x_i = 2 +4+6+8+...+100$$
$${\text{We know that,}}$$
$$\Rightarrow \bar {x} = \dfrac{\sum x_i}{n}$$

$$\Rightarrow \bar {x} = \dfrac{2 +4+6+8+...+100}{50}$$

$$\Rightarrow \bar {x} = \dfrac{50 \times 51}{50}$$ $$[\because \sum 2n = n(n+1)]$$

$$\Rightarrow \bar {x} = 51$$
$${\textbf{Step-2: Put the values in variance formula.}}$$
$$\Rightarrow \sigma^2 = \dfrac{1}{n} \sum x_i^2 - \bar{(x)}^2$$
$$= \dfrac{1}{50} (2^2 +4^2+6^2+8^2+...+100^2) - {(51)}^2$$
$$= \dfrac{1}{50} [({2.1})^2 +({2.2})^2+({2.3})^2+({2.4})^2+...+({2.50})^2] - {(51)}^2$$
$$= \dfrac{1}{50} 2^2[({1})^2 +({2})^2+({3})^2+({4})^2+...+({50})^2] - {(51)}^2 ...(1)$$
$${\text{But,}}$$
$$\Rightarrow ({1})^2 +({2})^2+({3})^2+({4})^2+...+({n})^2 = \dfrac{n(n+1)(2n+1)}{6}$$
$$\Rightarrow ({1})^2 +({2})^2+({3})^2+({4})^2+...+({50})^2 = \dfrac{50(50+1)(2 \times 50+1)}{6}$$
$$\Rightarrow ({1})^2 +({2})^2+({3})^2+({4})^2+...+({50})^2 = \dfrac{50(51)(101)}{6}$$
$${\text{Equation (1) become,}}$$
$$= \dfrac{1}{50} 2^2[\dfrac{50(51)(101)}{6}] - {(51)}^2$$
$$= 34 \times 101 -2601$$
$$ = 3434-2601$$
$$=833$$
$${\textbf{Thus, option B is correct.}}$$
Question 2
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The following line graph shows the yearly sales figures for a manufacturing company. Compute the difference between the sales (in Rs. crores) in $$2002$$ and $$2006$$.

A
$$4$$

B
$$6$$

C
$$7$$

D
$$10$$

Solution

From the graph the value on $$y $$ - axis (denoting sales) for the year $$2002$$ is Rs $$4$$ crore.
From the graph the value on $$y $$ - axis (denoting sales) for the year $$2006$$ is Rs $$8$$ crore.
Difference in sales $$= 8 - 4$$ = $$4$$ crore.
Question 3
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The following line graph shows the yearly sales figures for a manufacturing company. What were the sales (in Rs. crores) in $$2002$$?

A
$$2$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

As shown in the above graph, $$y$$ axis represents the sales(in crores) and $$x$$ axis represents the years.
So in the year $$2002$$ the sales were upto $$4$$ crore.
Question 4
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The following line graph shows the yearly sales figures for a manufacturing company. What were the sales in $$2005$$?

A
$$4$$

B
$$6$$

C
$$7$$

D
$$10$$

Solution

From the graph the value on $$y - axis$$ (denoting sales) for the year $$2005$$ is Rs. $$10$$ crore
Question 5
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_____ represents data that changes continuously over period of time.

A
Line graph

B
Pie graph

C
Bar graph

D
Histogram

Solution

The line graphs shows a change in data over time. A line graph is useful for displaying data or information that changes continuously over time. Another name for a line graph is a line chart.
Hence, option A is correct.
Question 6
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Directions For Questions

The following graph shows the temperature of a patient in a hospital, recorded every hour.

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The patient's temperature was same for two times during the period given. These two times were:

A
1 pm and 2 pm

B
2 pm and 3 pm

C
12 noon and 3 pm

D
9 am and 10 pm

Solution

From the graph the temperature curve is straight at 1 PM and at 2 PM.
This denotes that the temperature of the patient remained same from 1 PM to 2 PM
Hence, the answer is 1 pm and 2 pm.
Question 7
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The variance is the _______ of the standard deviation.

A
Square

B
Cube

C
Square root

D
Cube root

Solution

Standard Deviation:
The Standard Deviation is a measure of how spread out numbers are.
Its symbol is (the greek letter sigma)
The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?"
Variance:
The Variance is defined as: The average of the squared differences from the Mean.
Question 8
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The following line graph shows the yearly sales figures for a manufacturing company. What were the sales (in Rs. crores) in $$2006$$?

A
$$2$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

From the graph the value on $$y - axis$$ (denoting sales) for the year $$2006$$ is Rs $$8$$ crore
Question 9
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The following line graph shows the yearly sales figures for a manufacturing company. What were the sales (in Rs. crores) in $$2003$$?

A
$$6$$

B
$$7$$

C
$$8$$

D
$$10$$

Solution

From the graph the value on $$y - axis$$ (denoting sales) for the year $$2003$$ is Rs $$7$$ crore
Question 10
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Draw a graph for the following:
Side of square (in cm)
2
3
4
5
6
Area (in cm$$^2$$)
4
9
16
25
36
Is the graph linear?

A
Yes

B
No

C
Cannot be determined

D
None of these

Solution

Plotting the given points on the graph, we get a graph as shown in the figure above.
As seen in the graph, the points do not lie on the same straight line.

Hence, the graph is not linear.