Statistics

The correct answer is 0
According to given data


So for any x_i , (x_i – 62)² ≤ 140
⇒ x_i > 50 ∀ i = 1, 2, 3, …7
Therefore, no student is going to score less than 50.

The correct answer is 238


After omitting two wrong observations


Then,

The correct answer is (B) : 529
Mean = np = 16
Variance = npq = 8
and

Given, mean = np = α.
variance







So, the correct option is (C):

The correct answer is (C) : 43

∴

∴
Let a be the correct observation and b is the incorrect observation then a + b = 70
and


∴ Correct variance

The correct answer is (A) : 60

And

⇒ a² + b² = 25 …(ii)
From (i) and (ii) (a, b) = (3, 4) or (4, 3)
Now mean deviation about mean

⇒ 25M = 60.

Given that the incorrect mean (M) and variance (V) of the observations are 15 each:

1. First, calculate the incorrect sum of the observations using the formula: ΣX = M × n = 15 × 10 = 150

2. Correct the sum by replacing the incorrect observation (25) with the correct observation (15). Thus, the corrected sum is: ΣXcorrect = 150 - 25 + 15 = 140

3. Compute the correct mean: Mcorrect = ΣXcorrect / n = 140 / 10 = 14

4. Use the variance formula: V = (ΣX²/n) - (M²) to find the correct ΣX²:

Given V = 15, replace with the incorrect data:

15 = (ΣX² / 10) - (15²)

Solving for ΣX² gives: ΣX² = 15 × 10 + 225 = 375

5. Correct the ΣX² by replacing 25² with 15²:

ΣX²correct = 375 - 625 + 225 = 225

6. Calculate the correct variance:

Vcorrect = (ΣX²correct / n) - (Mcorrect)²

= (225 / 10) - 14² = 22.5 - 196 = 1.5

7. Finally, find the correct standard deviation, which is the square root of the correct variance:

SDcorrect = √1.5 ≈ 1.22

The correct standard deviation is approximately 1.22, which confirms it fits within the expected range: 2,2.

The correct answer is (A) : 0
Mean
Variance



D = 1 – 4(1 – 5n)2
= 40n – 7
D cannot be a perfect square as all perfect squares will be of the form of 4λ or 4λ + 1
So, a cannot be natural number
Therefore , Number of values = 0

The mean of the numbers is 5, and the variance is 10.

Step 1: Calculate the mean

The formula for the mean is: Substituting the given values: Simplify:

Step 2: Use the formula for variance

The formula for variance is: Substituting the given variance and mean : Simplify: Multiply through by 6:

Step 3: Solve for and

We are given: Use the identity: Substituting: The quadratic equation for and is: Solve using the quadratic formula: Thus, and (or vice versa).

Step 4: Calculate the mean deviation

The formula for mean deviation is: where is the mean. Substituting the values: Simplify:

Final Answer:

The mean deviation is:

Given: Intervals:

Frequencies: , and the mean .

Step 1: Calculate the mean:

The formula for the mean is:

,

where are the frequencies and are the midpoints of the intervals.

• Midpoints:
• Frequencies:

Substitute into the formula:

.

Simplify the numerator:

.

Simplify further:

.

Expand and solve for :

,

,

.

Step 2: Variance formula:

.

• Updated frequencies:
• Compute :

,

.

• Total frequency:

Substitute into the variance formula:

,

.

Final Answer: The variance of the data is .

Step 1: Use the mean to form an equation.
Step 2: Use the variance to form another equation.
Step 3: Solve the equations.
Expanding: Factoring: Since , and .
Step 4: Calculate .
Final Answer: .

The mean of the 7 observations is:

The total sum of the observations is:

When one observation is omitted, the sum of the remaining 6 observations is:

The mean of the remaining 6 observations is:

The variance of the 7 observations is given as . Using the formula for variance:

Substituting the known values:

Thus:

The variance of the remaining 6 observations is given by:

Substitute the known values:

We also know:

Substitute this value into the variance formula:

We now compute:

Simplify :

Thus:

Conclusion

The value of is:

The correct answer is option (C) : 178

Given Information:

• The mean of 7 observations is 8.
• The variance of 7 observations is 16.
• One observation, 14, is omitted.

We need to find the new mean a and variance b of the remaining 6 observations, and then calculate a + b.

Step 1: Calculate the Sum of the 7 Observations

The mean of 7 observations is given by:
μ = sum of observations / 7
Given that the mean is 8:
8 = sum of observations / 7
Thus, the sum of the observations is:
sum of observations = 8 × 7 = 56

Step 2: Calculate the Sum of Squared Deviations for the 7 Observations

The formula for variance is:
σ² = Σ(xᵢ - μ)² / 7
We are told that the variance is 16, so:
16 = Σ(xᵢ - 8)² / 7
Multiplying both sides by 7:
Σ(xᵢ - 8)² = 16 × 7 = 112
This represents the sum of squared deviations of the 7 observations from the mean.

Step 3: Omit the Observation 14

When the number 14 is omitted, we are left with 6 observations. We need to find the new sum of squared deviations and the new mean for these 6 observations.

New Mean (a):

After removing the observation 14, the new sum of the remaining 6 observations is:
new sum of observations = 56 - 14 = 42
The new mean of the remaining 6 observations is:
a = new sum of observations / 6 = 42 / 6 = 7

New Variance (b):

To calculate the new variance, we first subtract the squared deviation of 14 from the total sum of squared deviations. The squared deviation of 14 from the mean is:
(14 - 8)² = 6² = 36
So, the new sum of squared deviations for the remaining 6 observations is:
Σ(xᵢ - 8)² = 112 - 36 = 76

Now, the new variance b is:
b = 76 / 6 = 38 / 3 ≈ 12.67

Step 4: Calculate a + b

Now, we calculate a + b, where a = 7 and b = 38 / 3.
a + b = 7 + 38 / 3 = 21 / 3 + 38 / 3 = 59 / 3
This simplifies to approximately:
a + b ≈ 19.67

The problem is to find the value of using the given frequency distribution and information about the mean deviation around the median.

The age distribution given in the question is:

The total number of students is:

So, we have:

Next, we need to find the median age. The median is the middle value in a frequency distribution when the students are arranged in ascending order. To find the median, we need the cumulative frequency distribution:

The median corresponds to the 20th value since the total number of students is 40. From the cumulative distribution, we can see that the median age is 18.

Now, the mean deviation about the median is given by:

With a mean deviation of 1.25, we have:

Simplifying within the modulus:

Simplifying further:

Which simplifies to:

We now have two equations:

Subtract the first equation from the second:

Solving gives:

Substitute in :

Therefore, becomes:

Thus, the value of is 44.

To solve the problem of finding the variance of the random variable , which represents the number of rotten apples in a draw of two apples from a mixture of rotten and good apples, we need to proceed with the following steps:

Step 1: Determine probabilities.

• Total apples = 3 (rotten) + 15 (good) = 18 apples.
• Total ways to select 2 apples from 18 apples = ways.

Step 2: Establish the probability distribution of .

• : We choose 0 rotten apples (only good ones) from the draw. This is given by selecting 2 out of 15 good apples:
• : We choose 1 rotten and 1 good apple. This is:
• : We choose 2 rotten apples. This is given by selecting 2 out of the 3 rotten apples:

Step 3: Calculate the expected value .

Step 4: Calculate the expected value of , .

Step 5: Calculate the variance .

• Variance is given by

Thus, the variance of is , matching the given correct answer.

To solve this problem, we'll correct the observations and then find the new mean and variance. Given:
1. Original Mean ( ) = 12
2. Original Standard Deviation ( ) = 3
3. Incorrect observation = 10, Correct observation = 12
4. Total number of observations (n) = 15

Step 1: Calculate the original total sum of observations.


Step 2: Update the total sum with the correct observation.
The difference in observations is . Therefore,


Step 3: Calculate the new mean ( ).



Step 4: Calculate the original variance ( ).


Step 5: Calculate the corrected variance.
The corrected sum of the squares of the observations:\br
Removing the square of the incorrect observation and adding the square of the correct one:

Corrected variance:


Step 6: Calculate .


(rounded to the nearest integer is 2521)

Hence, the finally computed expression lies within the required range of 2521, confirming our calculations: 2521.

To find the standard deviation of the observations , we are given:

Let's use these conditions to find the standard deviation.

1. The formula for the variance of a data set is given by: where in this case.
2. We need to determine . We know:
   • Plugging in the given values:
   • Therefore:
   • Calculate :
3. Substitute into the variance formula:
4. The standard deviation is:

Hence, the standard deviation of the observations is , which matches the option given.

To solve the given problem, we need to decipher the information about the 10 observations in terms of their mean and variance, and how they relate to and .

1. We are given:

1. and

1. .
2. The mean of the observations is

1. . We know:

1. which gives

1. .
2. According to the formula for mean:

1. . Substituting for :

1. .
2. The variance of the observations is:

1. . For the variance:

1. , leading to:

1. .
2. We are also given:

1. . Hence:

1. , which simplifies to:

1. . Solving:

1. .
2. This yields:

1. . We conclude:

1. .
2. Finally, the ratio is:

1. .

Hence, the correct answer is: 2.

To solve this problem, we need to determine the values of and based on the given conditions, and then find the value of . Let's analyze the information step-by-step:

Step 1: Understanding the Mean

The mean of the observations is given as 55. The mean is calculated by the formula:

Substituting the given mean:

Multiplying through by 6 gives:

Thus,

(Equation 1)

Step 2: Understanding the Variance

The variance of the observations is given as 194. The formula for variance is:

Substituting the known values and simplifying:

Therefore, substituting these values in:

Simplifying further:

Multiplying through by 6 gives:

Thus,

(Equation 2)

Step 3: Solving the Equations

We have two equations:

Substitute into Equation 2:

55-b = \pm 20

Hence, or .

Because , we assign:

• If , then .
• If , then .

Thus, given , we take and .

Step 4: Calculation

Finally, compute :

The correct answer is 180.

To solve the given problem, we need to find the natural number such that the variance of the frequency distribution is 160. The distribution table is as follows:

Let's first find the mean, , of the distribution.

1. Calculate the total frequency: .
2. Calculate the sum of values: .
3. Calculate the mean: .

Given that the variance is 160, we will calculate it using the formula .

1. Calculate :
   • , , , , , .
   • So, .
2. Substitute into the variance formula and set it equal to 160:

1. Reorganize and simplify the equation:

1. Calculate \) :

Hence, the correct value of \) in natural numbers is 7. Therefore, the correct answer is 7.

To solve this problem, we need to use the concept of the median in a grouped frequency distribution. Given that:

• Median = 14
• Median Class Interval = 12-18
• Median Class Frequency = 12
• Number of students with marks less than 12 = 18

The formula for finding the median in a grouped data set is:

• is the lower limit of the median class (12 in this case).
• is the total number of observations.
• is the cumulative frequency of the class preceding the median class (students getting marks less than 12, which is 18).
• is the frequency of the median class (12).
• is the class width (18 - 12 = 6).

Substitute the known values into the formula:

Solving for :

Convert this to fraction and solve:

Therefore, the total number of students is 44. This matches with the given correct answer option.