Question

Two studies were completed in California. One study in northern California involved 1,000 patients; 74% of them experienced flulike symptoms during the month of December. The other study, in southern California, involved 500 patients; 34% of them experienced flulike symptoms during the same month. Which study has the smallest margin of error for a 98% confidence interval?
The northern California study with a margin of error of 3.2%.
The southern California study with a margin of error of 3.2%.
The northern California study with a margin of error of 4.9%.
The southern California study with a margin of error of 4.9%.

Answer 1

The northern California study with a margin of error of 3.2%.

Answer 2

Answer:

A. The northern California study with a margin of error of 3.2%.

Step-by-step explanation:

We know that,

$\text{M.E}=Z_{critical}\cdot \sqrt{\dfrac{p(1-p)}{n}}$

Where,

M.E = margin of error,

$Z_{critical}$ = z score of the confidence interval,

for 98% confidence interval $Z_{critical}=2.33$

p = proportion,

n = sample size.

One study in northern California involved 1,000 patients; 74% of them experienced flu like symptoms during the month of December.

Putting the values,

$\text{M.E}=2.33\cdot \sqrt{\dfrac{0.74(1-0.74)}{1000}}=0.032=3.2\%$

The other study, in southern California, involved 500 patients; 34% of them experienced flu like symptoms during the same month.

Putting the values,

$\text{M.E}=2.33\cdot \sqrt{\dfrac{0.34(1-0.34)}{500}}=0.049=4.9\%$

The smallest margin of error is 3.2% of the northern California study.