Compute the following probabilities: If Y is distributed N(1, 4), find Pr ( Y ≤ 3 ) . If Y is distributed N(3, 9), find Pr ( Y > 0 ) . If Y is distributed N(50, 25), find Pr ( 40 ≤ Y ≤ 52 ) . If Y is distributed N(5, 2), find Pr ( 6 ≤ Y ≤ 8 ) .

a) If Y is distributed N(1, 4), Pr (Y ≤ 3) = 0.84134

b) If Y is distributed N(3, 9), Pr (Y > 0) = 0.84134

c) If Y is distributed N(50, 25), Pr (40 ≤ Y ≤ 52) = 0.63267

d) If Y is distributed N(5, 2), find Pr (6 ≤ Y ≤ 8) = 0.22185

Step-by-step explanation:

With the logical assumption that all of these distributions are normal distribution,

a) Y is distributed N(1, 4), find Pr ( Y ≤ 3 )

Mean = μ = 1

Standard deviation = √(variance) = √4 = 2

To find the required probability, we first standardize 3

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (y - μ)/σ = (3 - 1)/2 = 1

We'll use data from the normal probability table for these probabilities

The required probability

Pr ( Y ≤ 3 ) = P(z ≤ 1) = 0.84134

b) If Y is distributed N(3, 9), find Pr ( Y > 0 )

Mean = μ = 3

Standard deviation = √(variance) = √9 = 3

To find the required probability, we first standardize 0

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (y - μ)/σ = (0 - 3)/3 = -1

We'll use data from the normal probability table for these probabilities

The required probability

Pr ( Y > 0) = P(z > -1) = 1 - P(z ≤ -1) = 1 - 0.15866 = 0.84134

c) If Y is distributed N(50, 25), find Pr (40 ≤ Y ≤ 52).

Mean = μ = 50

Standard deviation = √(variance) = √25 = 5

To find the required probability, we first standardize 40 and 52.

For 40,

z = (y - μ)/σ = (40 - 50)/5 = -2

For 52,

z = (y - μ)/σ = (52 - 50)/5 = 0.4

We'll use data from the normal probability table for these probabilities

The required probability

Pr (40 ≤ Y ≤ 52) = P(-2.00 ≤ z ≤ 0.40)

= P(z ≤ 0.40) - P(z ≤ -2.00)

= 0.65542 - 0.02275

= 0.63267

d) If Y is distributed N(5, 2), find Pr ( 6 ≤ Y ≤ 8 )

Mean = μ = 5

Standard deviation = √(variance) = √2 = 1.414

To find the required probability, we first standardize 6 and 8.

For 6,

z = (y - μ)/σ = (6 - 5)/1.414 = 0.71

For 8,

z = (y - μ)/σ = (8 - 5)/1.414 = 2.12

We'll use data from the normal probability table for these probabilities

The required probability

Pr (6 ≤ Y ≤ 8) = P(0.71 ≤ z ≤ 2.12)

= P(z ≤ 2.12) - P(z ≤ 0.71)

= 0.983 - 0.76115

= 0.22185

Hope this Helps!!

d 192

step-by-step explanation:

we find our area to find area we multiply thoses numbers to get our answer so our area is 192