If i have 37 test left and do 6 a day how many weeks will that take me to be done
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Answer:
a) The probability of Jack scoring higher is 0.3446
b) They probability of them scoring above 350 is 0.2119
Step-by-step explanation:
Lets call X the random variable that determines Jill's bowling score and Y the random variable that determines jack's. We have
Note that we are considering the variance on the second entry, the square of the standard deviation.
If we have two independent Normal distributed random variables, then their sum is also normally distributed. If fact, we have this formulas:
for independent distributions ,
, and a real number r.
a) We define Z to be Y-X. We want to know the probability of Z being greater than 0. We have
So Z is a normal random variable with mean equal to -10 and vriance equal to 625. The standard deviation of Z is √625 = 25.
Lets work with the standarization of Z, which we will call W. = (Z+10)/25. W has Normal distribution with mean 0 and standard deviation 1. We have
P(Z > 0) = P( (Z+10)/25 > (0+10)/25) = P(W > 0.4)
To calculate that, we will use the <em>known </em>values of the cummulative distribution function Φ of the standard normal distribution. For a real number k, P(W < k) = Φ(k). You can find those values in the Pdf I appended below.
Since Φ is a cummulative distribution function, we have P(W > 0.4) = 1- Φ(0.4)
That value of Φ(0.4) can be obtained by looking at the table, it is 0.6554. Therefore P(W > 0.4) = 1-0.6554 = 0.3446
As a result, The probability of Jack's score being higher is 0.3446. As you may expect, since Jack is expected to score less that Jill, the probability of him scoring higher is lesser than 0.5.
b) Now we define Z to be X+Y Since X and Y are independent Normal variables with mean 160 and 170 respectively, then Z has mean 330. And the variance of Z is equal to the sum of the variances of X and Y, that is, 625. Hence Z is Normally distributed with mean 330 and standard deviation rqual to 25 (the square root of 625).
We want to know the probability of Z being greater that 350, for that we standarized Z. We call W the standarization. W is s standard normal distributed random variable, and it is obtained from Z by removing its mean 330 and dividing by its standard deviation 25.
P(Z > 350) = P((Z - 330)/25 > (350-330)/25) = P(W > 0.8) = 1-Φ(0.8)
The last equality comes from the fact that Φ is a cummulative distribution function. The value of Φ(0.8) by looking at the table is 0.7881, therefore P(X+Y > 350) = 1 - Φ(0.8) = 0.2119.
As you may expect, this probability is pretty low because the mean value of the sum of their combined scores is quite below 350.
I hope this works for you!
The volume of the prop is calculated to be 2,712.96 cubic inches.
<u>Step-by-step explanation:</u>
Step 1:
The prop consists of a cone and a half-sphere on top. We will have to calculate the volumes of the cone and the half-sphere separately and then add them to obtain the total volume.
Step 2:
The volume of a cone is determined by multiplying with π, the square of the radius (r²) and height (h). Here we substitute π as 3.14. The radius is 9 inches and the height is 14 inches.
The volume of the cone : =
= 1,186.92 cubic inches.
Step 3:
The area of a half-sphere is half of a full sphere. The volume of a sphere is given by multiplying with π and the cube of the radius (r³).
Here the radius is 9 inches. We take π as 3.14.
The volume of a full sphere = =
= 3,052.08 cubic inches.
The volume of the half-sphere = = 1,526.04 cubic inches.
Step 4:
The total volume = The volume of the cone + The volume of the half sphere,
The total volume = 1,186.92 + 1,526.04 = 2,712.96 cubic inches.