P336. 7. The capacities ( in ampere-hours) were measured for a sample of 120 batteries. The average was 178 and the standard deviation was 14. (a) Find a 95% confidence interval for the mean capacity of batteries produced by this method. (b) Find a 99% confidence interval for the mean capacity of batteries produced by this method. (c) An engineer claims that the mean capacity is between 176 and 180 ampere-hours. With what level of confidence can this statement be made? (d) Approximately how many batteries must be sampled so that a 95% confidence interval will specify the mean to within ±2 ampere-hours? (e) Approximately how many batteries must be sampled so that a 99% confidence interval will specify the mean to within ±2 ampere-hours?

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

area = length x width

length = 2 x width

32 = 2 x width x width

32 = 2 x width^2

36 = width ^2  

width = 6

length = 12

area is 32

perimeter = 2 x 12 + 2 x 6 = 24 + 12 = 36

step-by-step explanation: