Normal Distribution for CUET Applied Maths

AnswerB · It is symmetric about its mean

1. The normal curve is bell-shaped and symmetric about the mean $\mu$, so its mean, median and mode coincide.

2. Its tails come ever closer to the axis but never meet it.

AnswerD · $1$

1. The area under the curve is the total probability.

2. Total probability is always 1, whatever $\mu$ and $\sigma$ are.

AnswerC · Both I and II

1. I: a normal variable can take any value in an interval, such as a height or a weight. True.

2. II: the peak of the bell is at the mean, which is also the mode. True.

AnswerC · $0.5$

1. The normal curve is symmetric about its mean.

2. So exactly half the area lies below the mean: 0.5.

AnswerA · Both A and R are true, and R is the correct explanation of A.

1. R is a defining property of the normal curve. True.

2. Symmetry splits the total area of 1 into two equal halves at $\mu$, so A is true because of R.

AnswerA · I, II and III only

1. Symmetry puts the mean, median and mode together (I) and half the area on each side of the mean (III); the tails only approach the axis (II).

2. IV is false: a standard deviation measures spread and is never negative.

AnswerC · B has the larger standard deviation

1. Both curves enclose an area of 1, so a wider curve must be lower.

2. B is lower and more spread out, so its standard deviation is larger.

AnswerB · They have the same shape, and B is A moved 20 to the right

1. The standard deviation sets the shape; equal standard deviations mean identical shapes.

2. The mean sets the position: B's centre is 20 units to the right of A's.

AnswerD · I, II and IV only

1. I: the total area stays 1, so a wider spread means a lower peak. II: the peak is at the mean. IV: a single point has no width, so no area.

2. III is false: the normal curve is symmetric, not skewed.

AnswerB · $1.5$

1. $z = \frac{x - \mu}{\sigma}$.

2. $z = \frac{72 - 60}{8} = \frac{12}{8} = 1.5$.

AnswerB · $-2$

1. $z = \frac{x - \mu}{\sigma}$.

2. $z = \frac{42 - 50}{4} = -2$.

AnswerC · $1.5$

1. $z = \frac{x - \mu}{\sigma}$.

2. $z = \frac{74 - 65}{6} = 1.5$.

AnswerC · $2$

1. $z = \frac{76 - 60}{8}$.

2. $\frac{16}{8} = 2$.

AnswerA · $48$

1. $x = \mu + z\sigma$.

2. $60 + (-1.5)(8) = 60 - 12 = 48$.

AnswerA · $44$

1. $x = \mu + z\sigma$.

2. $x = 50 + (-1.2)(5) = 50 - 6 = 44$.

AnswerD · Mean 0, standard deviation 1

1. $Z = \frac{X - \mu}{\sigma}$ shifts the mean to 0 and scales the spread to 1.

2. So $Z$ has mean 0 and standard deviation 1, whatever $X$ was.

AnswerB · Asha, with $z = 1.5$ against 1

1. Asha: $z = \frac{72 - 60}{8} = 1.5$. Ravi: $z = \frac{80 - 70}{10} = 1$.

2. Asha is 1.5 standard deviations above her class, Ravi only 1, so Asha did better.

AnswerC · $8$

1. $1.5 = \frac{92 - 80}{\sigma}$.

2. $\sigma = \frac{12}{1.5} = 8$.

AnswerD · $132$

1. $x = \mu + z\sigma$.

2. $120 + 0.8 \times 15 = 132$.

AnswerB · $\frac{X - \mu}{\sigma}$

1. Subtracting $\mu$ moves the mean to 0; dividing by $\sigma$ makes the standard deviation 1.

2. So $Z = \frac{X - \mu}{\sigma}$ has mean 0 and standard deviation 1.

AnswerD · Physics, since $z = 1.6$ against $z = 1.25$

1. Physics: $z = \frac{68 - 60}{5} = 1.6$. Chemistry: $z = \frac{75 - 65}{8} = 1.25$.

2. The higher $z$ is the better relative score: Physics.

AnswerB · Mathematics: his $z$-score is 2, against 1.5 in English

1. Mathematics: $z = \frac{76 - 60}{8} = 2$. English: $z = \frac{70 - 55}{10} = 1.5$.

2. A higher $z$-score means a better standing, so Mathematics.

AnswerA · 15.87%

1. $z = \frac{52 - 60}{8} = -1$.

2. $0.5 - 0.3413 = 0.1587$, so 15.87%.

AnswerD · $\mu = 50,\ \sigma = 10$

1. $65 = \mu + 1.5\sigma$ and $45 = \mu - 0.5\sigma$. Subtracting: $20 = 2\sigma$, so $\sigma = 10$.

2. Then $\mu = 45 + 5 = 50$.

AnswerA · Mean 50, standard deviation 10

1. $70 = \mu + 2\sigma$ and $40 = \mu - \sigma$.

2. Subtracting: $30 = 3\sigma$, so $\sigma = 10$ and $\mu = 50$.

AnswerB · 68%

1. The areas within 1, 2 and 3 standard deviations of the mean are about 68%, 95% and 99.7%.

2. So about 68% lies between $\mu - \sigma$ and $\mu + \sigma$.

AnswerD · $0.7881$

1. The left half has area 0.5, and 0.2881 lies between 0 and 0.8.

2. $0.5 + 0.2881 = 0.7881$.

AnswerC · $0.95$

1. By symmetry, the area from $-1.96$ to 0 is also 0.475.

2. $2 \times 0.475 = 0.95$.

AnswerB · $0.4772$

1. By symmetry, the area from $-2$ to 0 equals the area from 0 to 2.

2. So it is 0.4772.

AnswerB · 5%

1. About 95% lies within 2 standard deviations of the mean.

2. So about $100\% - 95\% = 5\%$ lies beyond, split between the two tails.

AnswerC · $1.5$

1. $z = \frac{x - \mu}{\sigma}$.

2. $\frac{165 - 150}{10} = 1.5$.

AnswerC · $0.9332$

1. The area to the left of 0 is 0.5, by symmetry.

2. $P(Z < 1.5) = 0.5 + 0.4332 = 0.9332$.

AnswerA · $0.0228$

1. The area to the right of 0 is 0.5, and 0.4772 of it lies between 0 and 2.

2. $P(Z > 2) = 0.5 - 0.4772 = 0.0228$.

AnswerC · $0.8185$

1. The shaded region straddles 0: by symmetry, the part from $-1$ to 0 equals the part from 0 to 1, 0.3413.

2. Shaded area $= 0.3413 + 0.4772 = 0.8185$.

AnswerA · $0.1587$

1. By symmetry, $P(Z < -1) = P(Z > 1)$.

2. $= 0.5 - 0.3413 = 0.1587$.

AnswerB · $0.0668$

1. The whole right half has area 0.5, and 0.4332 of it lies between 0 and 1.5.

2. Shaded tail $= 0.5 - 0.4332 = 0.0668$.

AnswerA · $0.3364$

1. Both z-values are right of 0: subtract the smaller area from the larger.

2. $0.4918 - 0.1554 = 0.3364$.

AnswerD · $0.8413$

1. The shaded area is everything to the left of $z = 1$: the left half, 0.5, plus 0.3413.

2. $0.5 + 0.3413 = 0.8413$.

AnswerA · $0.1151$

1. The right half has area 0.5, of which 0.3849 lies between 0 and 1.2.

2. $0.5 - 0.3849 = 0.1151$.

AnswerC · $0.6898$

1. The interval straddles 0: the part from $-0.6$ to 0 equals the part from 0 to 0.6, 0.2257.

2. $0.2257 + 0.4641 = 0.6898$.

AnswerB · 30.85%

1. $z = \frac{145 - 150}{10} = -0.5$.

2. $P(Z < -0.5) = 0.5 - 0.1915 = 0.3085$, so 30.85%.

AnswerD · 77.45%

1. $z = -1$ at 140 cm and $z = 1.5$ at 165 cm, on opposite sides of the mean.

2. $0.3413 + 0.4332 = 0.7745$, so 77.45%.

AnswerB · $0.1056$

1. By symmetry, $P(Z < -1.25) = P(Z > 1.25)$.

2. $0.5 - 0.3944 = 0.1056$.

AnswerD · $0.5987$

1. $P(Z > -0.25)$ is the whole right half plus the part from $-0.25$ to 0, which equals 0.0987.

2. $0.5 + 0.0987 = 0.5987$.

AnswerA · $0.2417$

1. Both z-values are on the right of 0, so the area between them is the larger area minus the smaller.

2. $0.4332 - 0.1915 = 0.2417$.

AnswerC · 169.6 cm

1. The top 2.5% lie above $z = 1.96$, since $0.5 - 0.4750 = 0.025$.

2. $x = 150 + 1.96 \times 10 = 169.6$ cm.