Statistical inference is based upon mathematical laws of probability. The following example will give you the basic ideas.
Suppose we want to determine if a certain coint is biased. We might do a few coin tosses (sample) so that we can decide if a particular coin is equally likely to land head or tail over an infinite number of tosses (population).
If we toss the coin ten times and get 6 heads and 4 tails, we might suspect the coin is biased towards heads, but we wouldn't be very confident about this, because it's not that unusual (not that improbable) to get 6 heads out of 10.
On the other hand, if we toss the coin ten times and get 10 heads - we would be more confident that the coin is biased towards heads, because it is very unusual (not very probable at all) that we would get this result from an unbiased coin.
The most common kind of statistical inference is hypothesis testing. Statistical data analysis allows us to use mathematical principles to decide how likely it is that our sample results match our hypothesis about a population. For example, if our research hypothesis is that the coin is not fair, but is actually biased towards heads - we can use principles of statistics to tell us how likely it is that we could get our sample results even if the coin were fair after all (null hypothesis).
If the probability of getting our sample results from a fair coin (for example) is very low, we feel confident in rejecting the null hypothesis (that the coin is fair). Even though we can't say for sure (because even a fair coin could produce our sample results), we can say that the results of our sample provide strong evidence against the null hypothesis, and we conclude that the coin is biased.
When we make this decision about a population based upon a sample, this is statistical inference.
In statistical hypothesis testing we use a p-value (probability value) to decide whether or not the sample provides strong evidence against the null hypothesis.
The p-value is a numerical measure of the statistical significance of a hypothesis test. It tells us how likely it is that we could have gotten our sample data (e.g., 10 heads) even if the null hypothesis is true (e.g., fair coin). By convention, if the p-value is less than 5% (p < 0.05), we conclude that the null hypothesis can be rejected (i.e., the coin is not fair). In other words, when p < 0.05 we say that the results are statistically significant, meaning we have strong evidence to suggest the null hypothesis is false.
Sample Size and Power Analysis are closely related to statistical inference. You might also look at the T-Test tutorial for another example of how statistical data analysis is used to make inferences from research data.
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