Some biologists believe the evolution of handedness is linked to complex behaviors such as tool-use. Under this theory, handedness would be genetically passed on from parents to children. That is, left-handed parents would be more likely to have lefthanded children than right-handed parents. An alternate theory asserts that handedness should be random, with left- and right-handedness equally likely. In a recent study using a simple random sample of n  76 right-handed parents, 50 of the children born were right-handed. ( pˆ =0.658 .) Suppose handedness is a random occurrence with either hand equally likely to be dominant, implying that the probability of a right-handed offspring is p =0.50. Required:
a. Show that it is reasonable to approximate the sampling distribution of p̂ using a normal distribution.
b. Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least p̂ =0.658 ?

a) Since both np >= 5 and n(1-p) >= 5,  it is reasonable to approximate the sampling distribution of p^‚ using a normal distribution.

b) 0.5 = 50% probability of observing a sample proportion of at 0.658.

Step-by-step explanation:

a. Show that it is reasonable to approximate the sampling distribution of p̂ using a normal distribution.

We need that:

np >= 5

n(1-p) >= 5

Sample of 76, 50 of the children were born were right-handed.

So

np = 76*0.658 >= 5

n(1-p) = 76*0.342 >= 5

Since both np >= 5 and n(1-p) >= 5,  it is reasonable to approximate the sampling distribution of p^‚ using a normal distribution.

b. Assuming left- and right-handed children are equally likely from right-handed parents, what is the probability of observing a sample proportion of at least p̂ =0.658 ?

This is the mean, so 50% below and 50% above.

0.5 = 50% probability of observing a sample proportion of at 0.658.

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

b and c

step-by-step explanation: