Choose the best answer. The distribution of grade point averages for a certain college is approximately Normal with a mean of 2.5 and a standard deviation of 0.6. Within which of the following intervals would we expect to find approximately 81.5% of all GPAs for students at this college?

(a) (0.7, 3.1)
(b) (1.3, 3.7)
(c) (1.9, 3.7)
(d) (1.9, 4.3)
(e) (0.7, 4.3)

Within (1.9 , 3.7) intervals we would expect to find approximately 81.5% of all GPA's for students at this college.

Step-by-step explanation:

We are given that the distribution of grade point averages for a certain college is approximately Normal with a mean of 2.5 and a standard deviation of 0.6.

Let X = distribution of grade point averages for a certain college

SO, X ~ Normal()

The z-score probability distribution for normal distribution is given by;

                    Z =  ~ N(0,1)

where, = population mean = 2.5

            = standard deviation = 0.6

Now, to find that within which interval we expect to find approximately 81.5% of all GPA's for students at this college, we will find respective probabilities for given intervals.

(a) P(0.7 < X < 3.1) = P(X < 3.1) - P(X 0.7)

    P(X < 3.1) = P( < ) = P(Z < 1) = 0.84134

    P(X 0.7) =  P( ) = P(Z -3) = 1 - P(Z < 3)

                                                     = 1 - 0.99865 = 0.00135

Therefore, P(0.7 < X < 3.1) = 0.84134 - 0.00135 = 0.8399

This means that in this interval we are able to find approximately 83.99% of all GPA's for students at this college.

(b) P(1.3 < X < 3.7) = P(X < 3.7) - P(X 1.3)

    P(X < 3.7) = P( < ) = P(Z < 2) = 0.97725

    P(X 1.3) =  P( ) = P(Z -2) = 1 - P(Z < 2)

                                                     = 1 - 0.97725 = 0.02275

Therefore, P(1.3 < X < 3.7) = 0.97725 - 0.02275 = 0.9545

This means that in this interval we are able to find approximately 95.45% of all GPA's for students at this college.

(c) P(1.9 < X < 3.7) = P(X < 3.7) - P(X 1.9)

    P(X < 3.7) = P( < ) = P(Z < 2) = 0.97725

    P(X 1.9) =  P( ) = P(Z -1) = 1 - P(Z < 1)

                                                     = 1 - 0.84134 = 0.15866

Therefore, P(1.9 < X < 3.7) = 0.97725 - 0.15866 = 0.8186

This means that in this interval we are able to find approximately 81.86% of all GPA's for students at this college.

(d) P(1.9 < X < 4.3) = P(X < 4.3) - P(X 1.9)

    P(X < 4.3) = P( < ) = P(Z < 3) = 0.99865

    P(X 1.9) = P( ) = P(Z -1) = 1 - P(Z < 1)

                                                     = 1 - 0.84134 = 0.15866

Therefore, P(1.9 < X < 4.3) = 0.99865 - 0.15866 = 0.8399

This means that in this interval we are able to find approximately 83.99% of all GPA's for students at this college.

(e) P(0.7 < X < 4.3) = P(X < 4.3) - P(X 0.7)

    P(X < 4.3) = P( < ) = P(Z < 3) = 0.99865

    P(X 0.7) =  P( ) = P(Z -3) = 1 - P(Z < 3)

                                                     = 1 - 0.99865 = 0.00135

Therefore, P(0.7 < X < 4.3) = 0.99865 - 0.00135 = 0.9973

This means that in this interval we are able to find approximately 99.73% of all GPA's for students at this college.

From all these it is clear that the interval (1.9 , 3.7) is expected to find approximately 81.5% of all GPA's for students at this college.

2 x^2 -3x+p

sum of roots = 3/2

let other root be x

3+ x = 3/2

x= 3/2 - 3 = -3/2

product of roots = p/2

3× -3/2 = p/2

-9 = p

⭐⭐

the correct is answer is a 9-sided polygon!