Find The Appropriate Values Of N Subscript 1 And N
A confidence interval is calculated for a sample mean by taking repeated samples from the same population and finding a margin of error based on the standard deviation. This is done with the intention of estimating an unknown population mean with a 95% probability. In order to determine if two sample means are significantly different, a test can be performed using the known 95% confidence intervals. When estimating the mean difference in matched data, the formula for sample size takes into account the standard deviation of the outcome. For example, if the sample mean age is to be estimated with 95% confidence and the population standard deviation is known to be 15 years, a sample size of 20 would be appropriate.
In the case of estimating ( (\mu_1 - \mu_2) ) with a sampling error of 3.1 and 95% confidence, and given that ( \sigma_1 \approx 19 ) and ( \sigma_2 \approx 18 ), the sample size (n) needed for both groups can be calculated using the formula:
[ n = \left(\frac{Z \cdot (\sigma_1^2 + \sigma_2^2)}{E^2}\right) ]
Where: Z = Z-score corresponding to the desired level of confidence (for 95% confidence, Z ≈ 1.96) ( \sigma_1 ) = Standard deviation of the first group ( \sigma_2 ) = Standard deviation of the second group E = Sampling error
By plugging in the known values, the sample size can be calculated. After doing this calculation, we find that the sample size for each group needed to estimate ( (\mu_1 - \mu_2) ) with a sampling error of 3.1 and 95% confidence is approximately 124.
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