JEE Mathematics Statistics MCQs Set 11

Answer: (d) a = 3, b = 4

Explanation:
1. Mean 6 gives \( a + b + 23 = 30 \), so \( a + b = 7 \).
2. Variance: \( (a^2 + b^2 + 64 + 25 + 100)/5 - 36 = 6.8 \), so \( a^2 + b^2 + 189 = 214 \) and \( a^2 + b^2 = 25 \).
3. With \( a + b = 7 \) and \( a^2 + b^2 = 25 \), ab = \( (49 - 25)/2 = 12 \), so \( {a, b} = {3, 4} \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (a) 1

Explanation:
1. Population \( B \) is population \( A \) with every observation increased by 50.
2. Adding a constant does not change the variance.
3. So \( V_A = V_B \) and the ratio is 1.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (c) \( \bar x+\frac{n+1}{2} \)

Explanation:
1. The total increases by \( 1 + 2 + ... + n = n(n+1)/2 \).
2. The mean increases by \( n(n+1)/(2n) = (n+1)/2 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (c) 57 kg

Explanation:
1. Total weight of seven students = \( 7 x 55 = 385 \) kg.
2. Total of the six known weights = \( 52 + 58 + 55 + 53 + 56 + 54 = 328 \).
3. Seventh weight = \( 385 - 328 = 57 \) kg.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) Mean and mode

Explanation:
1. Sorted data: 5, 8, 8, 10, 10, 11, 12, 14, 16, 16, 16, 18, 18, 20, 20. The median (8th value) is 14, and the mode is 16 (three times).
2. After correcting one 16 to 18: 16 appears twice and 18 appears three times, so the mode becomes 18. The sorted middle value is still 14, so the median is unchanged.
3. The total increases by 2, so the mean changes. Mean and mode change.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) x = y = z

Explanation:
1. Mean = \( 50/10 = 5 \).
2. Sorted: 1, 2, 3, 5, 5, 5, 6, 6, 8, 9; median = \( (5 + 5)/2 = 5 \).
3. Mode = 5 (appears three times). So \( x = y = z \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (c) \( \sqrt{10} \)

Explanation:
1. The second set is the first set shifted by 20.
2. Standard deviation does not change when a constant is added to every observation.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) \( 2\bar x+3 \)

Explanation:
1. Mean of \( (2x + 3) = 2 \) (mean of x) + 3.
2. So the new mean is 2 x-bar \( + 3 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) 10.1

Explanation:
1. There are 101 numbers; the mean is \( 1 + 50d \).
2. Deviations from the mean are \( |k - 50| d \) for \( k = 0 \) to 100; their sum is 2d(1 + 2 + ... + 50) = 2550d.
3. Mean deviation = \( 2550d/101 = 255 \), so \( d = 255 x 101/2550 = 10.1 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (a) 10

Explanation:
1. Mean = \( 2(1/3) + 3(1/2) + 11(1/6) = 4 \).
2. Mean of squares = \( 4(1/3) + 9(1/2) + 121(1/6) = 26 \).
3. Variance = \( 26 - 16 = 10 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) 52.25

Explanation:
1. Total marks = \( 40(50) + 35(60) + 45(55) + 42(45) = 2000 + 2100 + 2475 + 1890 = 8465 \).
2. Total students = 162.
3. Average = \( 8465/162 = 52.25 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) Group A is less variable than Group B because Group A's coefficient of variation is smaller.

Explanation:
1. Compare relative variability by the coefficient of variation = SD / mean \( x 100 \).
2. Group \( A \): 10/22 = 45.5%. Group \( B \): 12/23 = 52.2%.
3. Group \( A \) has the smaller coefficient of variation, so it is less variable.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) 38

Explanation:
1. Total of five numbers = 150; total of remaining four = 112.
2. Excluded number = \( 150 - 112 = 38 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) \( a\bar x+b \)

Explanation:
1. Mean of (ax + b) = a (mean of x) + b.
2. So it is a x-bar \( + b \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (a) 78.00

Explanation:
1. Corrected sum = \( 170 - 20 + 30 = 180 \).
2. Corrected sum of squares = \( 2830 - 400 + 900 = 3330 \).
3. Variance = \( 3330/15 - (180/15)^2 = 222 - 144 = 78 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) \( \frac{nM-x_n+x'}{n} \)

Explanation:
1. Original total = nM.
2. New total = nM \( - x_n + x' \).
3. New average = (nM - x_n + x')/n.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) 14

Explanation:
1. The numbers form an AP with common difference 7 and 7 terms; mean = 30.
2. Squared deviations: 441, 196, 49, 0, 49, 196, 441, total 1372; variance = 196.
3. Standard deviation = 14.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) y = 1.02x + 8.74

Explanation:
1. Slope b = \( (nS_{xy} - S_xS_y)/(nS_{xx} - S_x^{2}) \) = (34670 - 28600)/(22880 - 16900) = 6070/5980, about 1.02.
2. Means: x-bar = 13 and y-bar = 22.
3. Intercept = \( 22 - 1.02 x 13 = 8.74 \). So \( y = 1.02x + 8.74 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (c) \( \frac{3n+1}{4} \)

Explanation:
1. Weighted mean = sum of \( k(k^2 + k) \) / sum of \( (k^2 + k) \) = (sum of \( k^3 \) + sum of \( k^2)/(sum \) of \( k^2 \) + sum of \( k \)).
2. Numerator = \( n(n+1)(3n+1)(n+2)/12 \) and denominator = \( n(n+1)(n+2)/3 \).
3. Ratio = \( (3n + 1)/4 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) 39.95, 14.98

Explanation:
1. Correct total = \( 8000 - 50 + 40 = 7990 \), so mean = 39.95.
2. Original sum of squares = \( 200(225 + 1600) = 365000 \); corrected = \( 365000 - 2500 + 1600 = 364100 \).
3. Variance = \( 364100/200 - 39.95^2 = 1820.5 - 1596.0 = 224.5 \), so SD = 14.98.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) 169.18

Explanation:
1. Original total = \( 50 x 169 = 8450 \).
2. Correct total = \( 8450 - 134 + 143 = 8459 \).
3. Correct mean = \( 8459/50 = 169.18 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) 18

Explanation:
1. By the Cauchy-Schwarz inequality, \( (\sum x_i)^{2} \le n \sum x_i^{2} \).
2. \( 6400 \le 400 n \), so \( n \ge 16 \).
3. Among the options only 18 satisfies this.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (c) \( \frac{\bar x+10\alpha}{\alpha} \)

Explanation:
1. New mean = x-bar/alpha \( + 10 \).
2. Combine: (x-bar + 10 alpha)/alpha.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (d) \( \frac{n\bar X-x_2+\lambda}{n} \)

Explanation:
1. Original total = \( n \) X-bar.
2. New total = \( n \) X-bar \( - x_2 + \lambda \).
3. New mean = (\( n \) X-bar - x_2 + lambda)/n.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (b) 5

Explanation:
1. The number of sixes is binomial with \( n = 180 \) and \( p = 1/6 \).
2. Variance = npq = \( 180 x (1/6) x (5/6) = 25 \).
3. Standard deviation = 5.

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.

Answer: (a) 65

Explanation:
1. Total marks = \( 100 x 72 = 7200 \); boys' total = \( 70 x 75 = 5250 \).
2. Girls = 30 students with total \( 7200 - 5250 = 1950 \).
3. Girls' mean = \( 1950/30 = 65 \).

Teacher's Note:
1. Write down the definition (mean, variance, SD, mean deviation) before substituting values.
2. Check the answer with a small example or by recomputing the totals once.