Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.66 and standard deviation 0.87. (a) if a random sample of 25 specimens is selected, what is the probability that the sample average sediment density is at most 3.00? between 2.66 and 3.00? (round your answers to four decimal places.) at most 3.00 between 2.66 and 3.00 (b) how large a sample size would be required to ensure that the probability in part (a) is at least 0.99? (round your answer up to the nearest whole number.)

a) 0.9744; 0.4744; b) 36

Step-by-step explanation:

The formula for a z score of a mean of a sample is

For part a,

Our mean, μ, is 2.66 and our standard deviation, σ, is 0.87.  Our sample size, n, is 25.  We want P(X ≤ 3.00):

z = (3.00-2.66)/(0.87÷√25)

z = 0.34/(0.87÷5)

z = 0.34/0.174 = 1.95

Using a z chart, we see that the area to the left of this value is 0.9744.

For the second question,

We want P(2.66 ≤ X ≤ 3.00):

z = (2.66-2.66)/(0.87÷√25) = 0/0.174 = 0

z = (3.00-2.66)/(0.87÷√25) = 0.34/0.174 = 1.95

The area under the curve to the left of z = 0 is 0.5000; the area under the curve to the left of z = 1.95 is 0.9744.  This means the area between them is

0.9744-0.5000 = 0.4744.

For part b,

We want our probability to be at least 0.99.  Since we have the area under the curve to the left in a z table, we will subtract from 1:

1-0.99 = 0.01

In the z table, we see the cell with the value closest to 0.01 is 0.0099; this corresponds with a z score of -2.33:

Multiply both sides by 0.87:

0.87(-2.33) = ((0.34√n)/0.87)

-2.0271 = 0.34√n

Divide both sides by 0.34:

-2.0271/0.34 = 0.34√n/0.34

-5.9621 = √n

Square both sides:

(-5.9621)² = (√n)²

35.55 = n

This rounds to 36.

d. $180

step-by-step explanation:

if the dimensions of the room is 9ft by 12 ft, we have the area of the floor as,

so the area of the floor is 108 square feet, converting that into square yards we get,

the cost of carpet is $15 per square yard, therefore the cost of carpet for 12 square yards is,

hence, the cost of carpet is $180.

assuming the dimensions of the room be,

15 ft by 12 ft

so, the area of the floor of that room is,

we can convert 180 square feet into square yards, we know that,

1 sq ft = 0.111 sq yard

so, 180 square feet is,

so the area of floor is 20 square yards.

now, the cost of carpet is 15 dollars per square yard, therefore the cost of carpet for 20 square yards is,

so, it will cost $300.

113.42

i hope that is right.

step-by-step explanation: