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leva [86]
3 years ago
9

You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard devia

tion is approximately σ = 58.2 σ=58.2. You would like to be 99% confident that your estimate is within 1 of the true population mean. How large of a sample size is required? Do not round mid-calculation.
Mathematics
1 answer:
gtnhenbr [62]3 years ago
3 0

Answer:

n=(\frac{2.58(58.2)}{1})^2 =22546.82 \approx 22547

So the answer for this case would be n=22547 rounded up to the nearest integer

Step-by-step explanation:

Let's define some notation

\bar X represent the sample mean

\mu population mean (variable of interest)

\sigma=58.2 represent the population standard deviation

n represent the sample size  

ME =1 represent the margin of error desire

The margin of error is given by this formula:

ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}    (a)

And on this case we have that ME =+1 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

n=(\frac{z_{\alpha/2} \sigma}{ME})^2   (b)

The critical value for 99% of confidence interval now can be founded using the normal distribution. The significance would be \alpha=0.01 and the critical value z_{\alpha/2}=2.58, replacing into formula (b) we got:

n=(\frac{2.58(58.2)}{1})^2 =22546.82 \approx 22547

So the answer for this case would be n=22547 rounded up to the nearest integer

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