Normal Distribution

1.       Suppose X~N(4,9), the standard deviation of X is

• 2

• 3

• 4

• 9

Answer explanation

The number 9 is the variance of X.

Suppose X~N(4,9), probability P(X > 5.5) can be found using GC by keying in

• a)   Normalcdf(5.5,E99, 4, 9)

• b)       Normalcdf(5.5,E99, 2, 9)

• c)     Normalcdf(5.5,E99, 4, 3)

• d)       Normalcdf(5.5,E99, 4, 81)

Answer explanation

You need to remember to give standard deviation instead of variance in the GC keystrokes.

Tags

CCSS.HSS.ID.A.4

In Discrete Random Variables chapter, when X is a discrete random variable, to find P(X>=5), we have P(X>=5) = 1 - P(X<=4). Can we do so for if X were to be following a normal distribution?

• Yes

• No

Answer explanation

No, you cannot. If you do that, you will miss out removing the probability where X is between 4 and 5, since X is a continuous random variable now. You will need just key in normalcdf (5, E99, mean, standard deviation) to find the probability.

Tags

CCSS.HSS.ID.A.4

Let H be heights of students. Find the probability that the heights of three students is at least 4m. The expression should be

• P ( 3H > 4)

• P (3H >= 4)

• P (H1 + H2 + H3 > 4)

• P (H1 + H2 + H3 >= 4)

Answer explanation

There are three students, so there should be 3 variables. At least means including the number 4 as well.

Tags

CCSS.HSS.CP.B.7

The heights of students, H, follow a normal distribution with mean 1.1 and variance 0.4. Find the probability that the heights of three students is more than 4m. The probability is P(H1+H2+H3 > 4). What is the distribution of H1+H2+H3?

• N (3 x 1.1, 3 x 0.4)

• N (3 x 1.1, 9 x 0.4)

• N (3 x 1.1, 3 x 0.16)

• N (3 x 1.1,

  3 x sq rt (0.4) )

Answer explanation

0.4 is already the variance, not standard deviation. The format is N (mean, variance).

Var (H1+H2+H3) = Var(H1) + Var(H2) + Var(H3)

=3 Var(H)

Let X~N (4, 9) and Y~N(16, 25), where X and Y are independent variables.

Var (Y-X) is

• 25 - 9 = 16

• 25 + 9 = 34

• 5 - 3 = 2

• 5 + 3 = 8

Answer explanation

Variance only have addition. DO NOT subtract.

Chickens and turkeys are sold by weight. The masses, in kg, of the chickens and turkeys are modelled as having independent normal distributions with the following means and standard deviation.

Chicken C : mean 2.2, standard deviation 0.5

Turkey T : mean 10.5, standard deviation 2.1

Chickens and sold at $3 per kg and turkeys are sold at $5 per kg.

Which expression represents the cost of 3 chickens and 2 turkeys?

• 3C + 2T

• C1+C2+C3+T1+T2

• 3(3C)+5(2T)

• 3(C1+C2+C3)+5(T1+T2)

Answer explanation

There are 5 animals, thus there should be 5 variables. The cost is multiplied to the total weight of the type of farm animals.

The masses, C, in kg, of the chickens is modelled as a normal distributions with the following mean and standard deviation.

Chicken C: mean 2.2, standard deviation 0.5

Chickens and sold at $4 per kg.

To find the cost of 3 randomly chosen chickens, Q, we need to find the variance of Q.

Which of the following is the variance of Q?

• (4)(3)(0.5)

• (4)(9)(0.25)

• (16)(3)(0.25)

• (16)(9)(0.25)

Answer explanation

Q=4(C1+C2+C3)

Var (Q) = 16Var (C1+C2+C3) = 16 (3) Var (C) = (16)(3)(0.25)

You need to square the standard deviation for variance.

Let C denote the mass of chicken and T denote the mass of turkey. Find the probability that the difference in mass of 2 chickens and 1 turkey is less than 3 kg is

• P ( C1 + C2 - T < 3)

• P( T - C1 - C2 < 3)

• P( | C1 + C2 - T | < 3)

• P( | C1 + C2 - T < 3| )

Answer explanation

You do not know whether the mass of 2 chickens or 1 turkey is bigger. Thus need to put modulus. Modulus should not include the value 3.

Let C denote the mass of chicken and T denote the mass of turkey. Find the probability that 4 chickens weigh more than a turkey by less than 2 kg. Which is the expression that best describe this probability?

• P ( C1 + C2 + C3 + C4 - T < 2 )

• P ( T - C1 - C2 - C3 - C4 < 2 )

• P ( | T - C1 - C2 - C3 - C4 | < 2 )

• P ( 0 < C1 + C2 + C3 + C4 - T < 2 )

Answer explanation

There must be a lower bound zero so that T is always more than mass of 4 chickens

Tags

CCSS.HSS.ID.A.4