Question

Hi, i would like to request assistance with the following question:

Bert computes a 95% confidence interval for p and obtains the interval [0.600, 0.700]. Note: Parts (a) and (b) are not connected: Part (b) can be answered even if one does not know how to do part (a). (a) Bert’s boss says, “Give me a 90% confidence interval for p.” Calculate the answer for Bert. (b) Bert’s boss says, “Give me a 95% confidence interval for p−q.” Calculate the answer for Bert. (Hint: p−q = p−(1−p) = 2p − 1. Bert’s interval says, in part, that “p is at least 0.600;” what does this tell us about 2p − 1?)

Answer 1

a) From standard normal tables, we have here:
P(-1.96 < Z < 1.96) = 0.95

And, P(-1.645 < Z < 1.645) = 0.9

Now we are given the 95% confidence interval for p as: [0.600, 0.700].

Therefore, the point estimate of proportion and the margin of error here are obtained as:
p = (0.6 + 0.7)/2 = 0.650
MOE = U - p = 0.7 - 0.65 = 0.05

The new margin of error is therefore computed now as:

= MOE *( New critical Z value / Old critical Z value)

= 0.05*( 1.645 / 1.96) = 0.0420

Therefore the confidence interval here is given as:

0.65 - 0.042, 0.65 + 0.042

= (0.608, 0.692)
This is the required 90% confidence interval for p here.

b) We have the 95% confidence interval for p here as:
[0.600, 0.700].

Therefore, 0.6 < p < 0.7

1.2 < 2p < 1.4

1.2 - 1 < 2p - 1 < 1.4 - 1

0.2 < 2p - 1 < 0.4

0.2 < p - q < 0.4

Therefore [0.200, 0.400] is the required confidence interval here.

p is at least 0.6 means that p >= 0.6 which means 2p - 1 >= 2*0.6 - 1 which means that 2p - 1 >= 0.2