Question

Two polls were conducted about the upcoming election for new governor. The first poll said that 45% of the people asked would vote for Mr. Johnson with a margin of error of plus or minus 4.6%. The second poll said that 42% of the people asked would vote for Mr. Johnson with a margin of error of plus or minus 3.6%.

Can the conclusions of both polls be correct?
a. yes, the two corresponding confidence intervals overlap.
b. no, the difference between 45% and 42% is too large.
c. no, the two corresponding confidence intervals do not overlap.

Answer 1

A. Yes, the two corresponding confidence intervals overlap.

Explanation:

The correct option is - A. Yes, the two corresponding confidence intervals overlap.

Reason -

P(1) = 45% = 0.45

E(1) = 4.6% = 0.046

Confidence Interval = P(1) ± E(1)

= 0.45 ± 0.046

So, the interval is -

0.45 + 0.046 ≤ P ≤ 0.45 - 0.046

⇒0.496 ≤ P ≤ 0.404 .............(1)

Now,

P(2) = 42% = 0.42

E(2) = 3.6 % = 0.036

Confidence Interval = P(2) ± E(2)

= 0.42 ± 0.036

So, the interval is -

0.42 + 0.036 ≤ P ≤ 0.42 - 0.036

⇒0.456 ≤ P ≤ 0.384 ...............(2)

From equation (1) and (2), we get

The two corresponding confidence intervals overlap.