A random sample of 144 observations produced a sample proportion of 0.4. An approximate 90% confidence interval for the population proportion p is between: a) 0.320 and 0.480
b) 0.333 and 0.480
c) 0.333 and 0.467
d) 0.359 and 0.441
e) 0.313 and 0.487

CI =(0.333, 0.480)

Step-by-step explanation:

The formula for calculating the confidence interval is expressed as shown;

CI = p±Z * √p(1-p)/n±0.5/n

Z is the z-score at 90% confidence

p is the sample proportion

n is the sample size

Given n = 144, p = 0.4 and z-score at 90%  CI = 1.645 (from z table)

Substituting this values;

CI = p ± 1.645*√0.4(1-0.4)/144 ±0.5/n

CI = 0.4 ± 1.645*√0.4(0.6)/144 ± 0.5/144

CI = 0.4 ±1.645 * √0.24/144 ± 0.00347

CI = 0.4 ±1.645 * 0.04087± 0.00347

CI = 0.4±0.06723±0.00347

CI =(0.333, 0.480)

d. bc = 12 and ef = 16

step-by-step explanation:

we are given that δabc : δdef such that .

also, it is given that

so, from the options, we need that .

a.

b.

c.  

d.

as, we can see that only option d gives the same ratio as i.e. .

hence, the possible lengths are bc = 12 and ef = 16.