Question

A study is going to be conducted in which a mean of a lifetime of batteries produced by a certain method will be estimated using a 90% confidence interval. The estimate needs to be within +/- 2 hours of the actual population mean. The population standard deviation s is estimated to be around 25. The necessary sample size should be at least _______.

Answer 1

Answer:

The necessary sample size should be at least 423.

Step-by-step explanation:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1-0.9}{2} = 0.05$

Now, we have to find z in the Ztable as such z has a pvalue of $1-\alpha$.

So it is z with a pvalue of $1-0.05 = 0.95$, so $z = 1.645$

Now, find the margin of error M as such

$M = z*\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the length of the sample.

In this problem, we have that:

$M = 2, \sigma = 25$. So

$2 = 1.645*\frac{25}{\sqrt{n}}$

$2\sqrt{n} = 41.125$

$\sqrt{n} = 20.5625$

$\sqrt{n}^{2} = (20.5625)^{2}$

$n = 422.81$

The necessary sample size should be at least 423.