Find the required sample size for the investigator.

Question

An investigator interested in estimating a population mean wants to be 95% certain
that the length of the confidence interval does not exceed 4. Find the required
sample size for his study if the population standard deviation is 14.

My work:

Confidence interval: $95$%

Error estimate(E): $4$

$\sigma=14$

$\alpha=0.05$ so $\alpha/2=0.025$

$Z_{0.025}=1.96$

so using $n =(\cfrac{Z_{0.025} \cdot \sigma}{E})^2 = (\cfrac{1.96 \cdot 14}{4})^2 = 47.0596$

The sample size cannot be a decimal so round up to $48$ so $n = 48$

So he needs a sample size of $48$ but the book's answer says it's $n = 189$

I have no idea how this is right. Please help me.

Answer 1

When the length of the confidence interval is 4, the value of E is 2.