Math 117 Practice Midterm 2

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

Problem 1

The following table gives the probability distribution for the number of kittens in a litter.

 Number x in Litter 1 2 3 4 5 6 P(x) 0.05 0.12 0.24 0.35 0.2 0.04

Determine E(x) and s.  Explain the meaning of each.

Solution

We expand the table as follows

 Number x in Litter 1 2 3 4 5 6 Totals P(x) 0.05 0.12 0.24 0.35 0.2 0.04 xP(x) 0.05 0.24 0.72 1.4 1 0.24 3.65 (x - 3.65)2 7.02 2.72 0.42 0.12 1.82 5.52 (x - 3.65)2(P(x)) 0.35 0.33 0.1 0.04 0.36 0.22 1.40

The expected value is 3.65.  This means that the overall average number of kittens in a litter is 3.65 kittens.

The variance is 1.40.  We take the square root to get the standard deviation of 1.18.

Problem 2

You have experimented with the days for germination for a lupin and know that the distribution follows the parabolic distribution

f(t)  =  kt(1 - t)        for         0  <  t  < 1

where t is measured in months and k is a positive constant.

A.  Find k.

Solution

To find k we set the integral equal to zero.

Hence k  =  6.

B.  Find the probability that a lupine seed will germinate between 0.3 and 0.5 months.

Solution

We find

Hence

P(0.3 < x < 0.5)  =  0.284

C.  Find the expected value E(t) and interpret what it means.

Solution

We find

Hence

E(t)  =  0.5

D.  Find the standard deviation.

Solution

We calculate the integral

Now calculate the variance

V  =  0.3 - 0.25  =  0.05

Take a square root to get the standard deviation of 0.22.

Problem 3

Currently, the benefit package costs the college \$9442 and is expected to increase 20% next year.  Assume the 20% increase becomes a trend over the next several years.

A.  Write an expression for the cost of the benefit package after n years.

Solution

Notice that a 20% increase means that the next year will have 1.2 times the current year, since you have 100% of your expense from last year plus an additional 20%.  Look at the first few terms.

a0  =  9442        a1  =  (9442)(1.2)

a2  =  (9442)(1.2)2         a3  =  (9442)(1.2)3

Hopefully the pattern is apparent.  The general term is

an  =  (9442)(1.2)n

B.  Compute the cost of benefits in the year 2020.  (This year is 2004).

Solution

The year 2020 corresponds with n = 2020 - 2004  =  16

a16  =  (9442)(1.2)16 = 174568

We can expect that the benefit package will cost \$174,568 if trends continue.

Problem 4

Determine the convergence or divergence of the following series.

A.

Solution

We take the limit using L'Hopital's rule:

Since the limit is not zero, the series diverges.

B.

Solution

This is a geometric series with r  =  1/3, hence by the geometric series test the series converges.

C.

S

Problem 5

Explain why the series converges.  Then approximate the following sum using 5 terms.  Then estimate the maximum error of your approximation.

Solution

This is a p-series with p  =  8 > 1, hence by the p-series test the series converges.

We find

We compute the maximum error by finding