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

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

Step-by-step explanation:

The formula for margin of error for proportion is given by :-

$z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}$

Given : Significance level : $\alpha:1-0.98=0.02$

Critical value : $z_{\alpha/2}=2.33$

For northern California , the sample size was takes : n= 1000

The proportion of patients  experienced flu like symptoms during the month of December = 0.74

Then , the margin of error will be :-

$(2.33)\sqrt{\dfrac{0.74(1-0.74)}{1000}}\approx0.032=3.2\%$

For southern California , the sample size was takes : n= 500

The proportion of patients  experienced flu like symptoms during the month of December = 0.34

Then , the margin of error will be :-

$(2.33)\sqrt{\dfrac{0.34(1-0.34)}{500}}\approx0.049=4.9\%$

Clearly, the northern California study with a margin of error of 3.2%  has the smallest margin of error for a 98% confidence interval.

Answer 2

The  study has the smallest margin of error for a 98% confidence interval is The northern California study with a margin of error of 3.2%.

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