Test: Statistics- 2 - JEE MCQ

The mean of the squares of the first n natural numbers is given by:

The sum of the squares of the first n natural numbers is:

Thus, the mean is:

We are given that the mean is 11. Therefore:

Multiply through by 6:
(n+1)(2n+1)=66
Expand:
2n²  +3n+1=66
Simplify:
2n²+3n−65=0
Solve this quadratic equation using the quadratic formula:

Here,  a=2, b=3, and  c=−65:

Calculate the two solutions:

Since n must be a positive integer, n=5.

The first  n odd natural numbers are 1,3,5,…,(2n−1). The mean of these numbers is calculated as:

The sum of the first n odd natural numbers is:
Sum=n²
So, the mean becomes:

We are given that the mean is equal to n itself. This equality holds true for any natural number n.

The given arithmetic progression (AP) is:
(a+d),(a+3d),(a+5d),…
This sequence has a common difference of 2d.
Step 1: General term
The n-th term of this AP is given by:
T_n =a+(2n−1)d
Step 2: Sum of the first n terms
The sum of the first n terms of an AP is:

Here: First term = a+d
Last term = T_n ​ =a+(2n−1)d

Simplify the terms:

Step 3: Mean of the first n terms
The mean of the first n terms is:

Substitute S_n​:

To find the median, we first arrange the data in ascending order (already done here):
13,14,16,18,20,22
Step 1: Count the number of observations
The number of data points is n = 6 (even).
Step 2: Median for even n
For an even number of observations, the median is the average of the two middle values. The two middle values here are 16 and 18.

If the two lines of regression are at right angles, it means that the product of their slopes is −1. This happens only when the correlation coefficient ρ(X,Y) is 0.
The correlation coefficient measures the linear relationship between two variables. Whe ρ(X,Y)=0, the two lines of regression are perpendicular, indicating no linear correlation.

If COV(X,Y)=0, it implies that the covariance between X and  Y is zero. This means that there is no linear relationship between the two variables. In such a case, the correlation coefficient  ρ(X,Y)=0, and the two lines of regression will be at right angles to each other.

If the variance of the data is denoted by V, then the standard deviation (S.D.) is the square root of the variance.
Mathematically,
S.D. = √V
The correct answer is: a) √V

To determine the correlation coefficient  ρ(X,Y) from the two regression lines, we use the property that the product of the slopes of the two regression lines is equal to  ρ ² .
Given regression lines:
y=3x−5
Slope (m₁ ​) = 3

y=2x−4
Slope ( m₂​) = 2
Step 1: Check the product of slopes
The product of the slopes of the regression lines is:
m ₁ ​ ⋅m ₂ ​ =3⋅2=6
However, for the two regression lines, the product of their slopes must satisfy:
m₁ ​ ⋅m₂ ​ =ρ 2
Thus: ρ² =6
Step 2: Conclusion
Since ρ² cannot exceed 1, this configuration of regression lines is not possible. Hence, ρ(X,Y) is undefined for the given regression lines.

We are given the relationship:
Median=(Mode+2⋅Mean)μ
Step 1: Recall the empirical relationship
The empirical relationship between the mode, mean, and median is:
Mode=3⋅Median−2⋅Mean
Step 2: Substituting the given formula
Rearrange the given formula to solve for μ:

From the empirical relationship:
Mode+2⋅Mean=3⋅Median
Substituting into the equation for μ:

We are given the relationship:
Mean=(3⋅Median−Mode)⋅x
Step 1: Recall the empirical relationship
The empirical relationship between the mean, median, and mode is:
Mode=3⋅Median−2⋅Mean
Step 2: Compare the given formula with the empirical relationship
Rewriting the empirical formula for the mean:

Compare this with the given formula:
Mean=(3⋅Median−Mode)⋅x
Equating the two expressions for the mean:

Step 3: Solve for x
Cancel 
3⋅Median−Mode (non-zero):

The least square lines of regression provide the best possible estimates when the correlation coefficient  ρ(X,Y) is equal to 1 or -1. This is because a correlation of ρ=1 or  ρ=−1 indicates a perfect linear relationship between X and Y, meaning one variable can be predicted exactly from the other using the regression line.

The quartile deviation (also called the semi-interquartile range) is approximately related to the standard deviation (σ) by the formula:

This relationship holds in cases where the data follows a normal distribution, as the quartile deviation is a measure of spread that focuses on the interquartile range.

In a symmetrical distribution, the median lies exactly halfway between the first quartile (Q₁) and the third quartile (Q₃). The formula for the median in such cases is:

Given:
Q_1​=20andQ₃​=40

A measure of dispersion quantifies the spread or variability of a dataset. Common measures of dispersion include:

• Mean deviation: Measures the average distance of data points from the mean or median.
• Variance: Measures the average squared deviation from the mean.
• Range: Measures the difference between the maximum and minimum values.

The mean, however, is a measure of central tendency, not dispersion.

We are given the data: 1, 2, 3, 4, 5, 6

Step 1: Calculate the Mean

The formula for the mean is:

Mean = (Sum of all values) / (Number of values)

Substituting the values:

Mean = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 7/2

Step 2: Calculate the Variance

The formula for variance is:

Variance = Σ(x_i - Mean)² / n

1. Subtract the mean (7/2) from each value and square the result:

• (1 - 7/2)² = (-5/2)² = 25/4
• (2 - 7/2)² = (-3/2)² = 9/4
• (3 - 7/2)² = (-1/2)² = 1/4
• (4 - 7/2)² = (1/2)² = 1/4
• (5 - 7/2)² = (3/2)² = 9/4
• (6 - 7/2)² = (5/2)² = 25/4

2. Sum the squared values:

(25/4) + (9/4) + (1/4) + (1/4) + (9/4) + (25/4) = 70/4 = 35/2

3. Divide by n (6):

Variance = (35/2) / 6 = 35/12

Step 3: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

Standard Deviation = √(35/12)

Final Answer

The mean is 7/2, and the standard deviation is √(35/12).

Correct option: (c) 7/2, √(35/12)

Step 1: Definition of Range
The range of a dataset is the difference between the maximum and minimum values.
Step 2: Identify Maximum and Minimum Values
From the given dataset:
2,3,5,9,8,7,6,5,7,4,3

• Maximum value: 9
• Minimum value: 2

Step 3: Calculate the Range
Range=Maximum−Minimum=9−2=7

The median is the middle value of a dataset when arranged in ascending order. It is also represented as:

• The 5th decile (because the median divides the data into two equal parts, which is the 5th decile in the decile system).
• The 2nd quartile (as quartiles divide the data into four equal parts, and the median is at the 2nd quartile).
• The 50th percentile (since percentiles divide the data into 100 equal parts, and the median corresponds to the 50th percentile).

The lower quartile (Q₁) is the first quartile, representing the 25th percentile, and is not equal to the median.

To calculate the mean deviation from the mean, follow these steps:
Step 1: Find the Mean
The observations are: 
−1,0,4.
The mean (xˉ ) is calculated as:

Step 2: Calculate the Absolute Deviations
Find the absolute differences between each observation and the mean:
∣−1−1∣=2,∣0−1∣=1,∣4−1∣=3
Step 3: Find the Mean Deviation
The mean deviation from the mean is:

Step 1: Calculate the total sum of the observations
The mean of 50 observations is 36. Therefore, the total sum of the 50 observations is: Total Sum = Mean × Number of Observations
Total Sum=Mean×Number of Observations
Total Sum = 36 × 50 = 1800 Total Sum=36×50=1800
Step 2: Subtract the deleted observations
Two observations, 30 and 42, are deleted. The sum of these two observations is: Sum of deleted observations = 30 + 42 = 72
Sum of deleted observations=30+42=72
The new total sum of the remaining observations is:
New Total Sum = 1800 − 72 = 1728 New Total Sum=1800−72=1728
Step 3: Calculate the mean of the remaining observations
The number of remaining observations is:
50 − 2 = 48 50−2=48
The new mean is:
50−2=48
The new mean is:

The regression coefficient b _yx ​ is defined as the change in Y per unit change in X. It is related to the correlation coefficient ρ(X,Y) and the regression coefficient b xy ​ through the relationship:
b_yx​⋅b_xy​=ρ²
Where ρ is the correlation coefficient, and ρ² lies between 0 and 1.
Key Consideration: If  b_yx ​ >1, it means that the slope of the regression line predicting Y from X is greater than 1. However, b yx ​ can still be positive or negative depending on the direction of the relationship.
The condition  b _yx ​ >1 implies a steeper slope of the regression line for Y on X. It does not necessarily indicate negativity or positivity, as it depends on the data.

To find the average speed of the whole journey, use the formula for the average speed in a round trip:

Where: v₁ ​ is the speed for the first part of the journey ( 20 km/h 20km/h),
v ₂ ​ is the speed for the return journey ( 30 km/h 30km/h).
Step 1: Substitute the values

To solve this, let the number of boys be b and the number of girls be g.
Step 1: Use the formula for the combined mean
The formula for the mean of a combined group is:

Given: Mean weight of boys =  80kg,
Mean weight of girls =  50kg,
Combined mean = 60kg.
The total weight of boys is 80b, and the total weight of girls is  50g. The total number of students is  b+g. Substituting these into the formula:

Step 2: Solve for the ratio b/g
Multiply through by  b+g to eliminate the denominator:
60(b+g)=80b+50g
Expand and simplify:
60b+60g=80b+50g
60g−50g=80b−60b
10g=20b

Step 3: Conclusion
The ratio of the number of boys to the number of girls is:
1:2

To find the value of x, we use the formula for the mean:

Step 1: Write the equation for the mean
Given:

• Observations: 3,4,x,7,10
• Mean: 6

Substitute into the formula:

Step 2: Simplify the equation

Multiply through by 5:
30=24+x
Solve for x:
x=30−24=6

For a normal distribution, the following property holds:
Mean=Median=Mode
This is because a normal distribution is symmetric about its center, and the mean, median, and mode all lie at the same central point.