# Notes on Inferential Statistics Rahul's Noteblog Notes on Biostatistics Notes on Inferential Statistics

## Inference:

The task of using s sample to draw conclusions about a population involves going beyond the actual information that is available.

## Sampling error:

This is not an error, but a natural, expected random variation that'll cause the sample statistic to differ from the population parameter.

## Definition of Central Limit Theorem:

This theorem states that the random sampling distribution of means will always tend to be normal, irrespective of the shape of the population. The theorem further states that the random sampling distribution of means will become closer to normal as the size of the samples increases.

## Standard Error:

• Standard error is the population standard deviation divided by the square-root of sample size.

• The more the sampling size, the less the standard error.

• The shape of the random sampling distribution of means, as reflected by its standard error is affected by size of the sample.

• This is why results of large studies are more trusted that those of small studies.

## Z score:

• Z-score is = (sample mean - population mean) divided by population standard deviation.

• The statistic Z-score tells us how far the observed sample mean is from the population mean.

• We can use Z-scores to calculate the probability that a sample will have a mean of above or below a given value.

## Confidence Limits:

• Confidence limit = sample mean + or - (Z score times sample standard deviation).

• Researchers place special emphasis on + and - 1.96 standard errors.

• Logically, if sample mean lies within + and - 1.96 standard errors of the population mean 95% of the time, then population mean must also lie + or - 1.96 standard errors of sample mean 95% of the time.

• The exact z score for a 95% confidence limit is 1.96.

• 68% confidence limits = Sample mean + or - approximately 1x standard error.

• 95% confidence limits = Sample mean + or - approximately 2x standard error.

• 99.7% confidence limits = Sample mean + or - approximately 3x standard error.

## Precision vs. Accuracy:

• Precision is the degree to which a figure (such as an estimate of a population mean) is immune from random variation. The width of the confidence interval (C.I.) reflects precision -- the wider the C.I. the less precise the estimate. Larger samples create smaller C.I.

• Accuracy is the degree to which an estimate is immune from systematic error or bias.

• Halving the width of the confidence interval is the same as doubling precision.

• When standard error is halved, width of confidence interval is also halved.

• Less standard deviation = more precise.

## Estimated Standard Error of Mean:

• Estimated Standard Error of Mean = Sample standard deviation divided by square-root of sample size.

• The above equation is used to calculate standard when population standard deviation is not known, instead, the sample standard deviation is known.

## T-scores:

• The estimated standard error is used to find a statistic t which is sometimes known as the student's t. This is the number of estimated standard errors by which the sample mean lies above or below the population mean.

• T score = (sample mean - population mean) / Estimated Standard Error of Mean.