Question

A sample has a sample proportion of 0.3. Which sample size will produce the widest 95% confidence interval when estimating the population parameter?A. 36B. 56C. 68D. 46

Answer 1

In this case, we'll have to carry out several steps to find the solution.

Step 01:

Data

sample proportion = 0.3

widest 95% confidence interval

sample = ?

Step 02:

p = 0.3

1 - α = 0.95 =>> z α/2 = 1.96

We must check each value to find the solution.

A. sample = 36

`\begin{gathered} 0.3-1.96\cdot\sqrt[]{(0.3\cdot0.7)/(36)}=0.3-0.1499 \\ 0.3+1.96\cdot\sqrt[]{(0.3\cdot0.7)/(36)}=0.3+0.1499 \end{gathered}`

confidence interval (0.1501 , 0.4499)

difference = 0.2998

B. sample = 56

`\begin{gathered} 0.3-1.96\cdot\sqrt[]{(0.3\cdot0.7)/(56)}=0.3\text{ - }0.120 \\ 0.3+1.96\cdot\sqrt[]{(0.3\cdot0.7)/(56)}=\text{ 0.3 + }0.120 \end{gathered}`

confidence interval (0.18 , 0.42)

difference = 0.24

Analyzing these two values, we can conclude that the widest confidence interval will be for the smallest sample.

The answer is:

Sample = 36