Generally, researchers are interested in the population means. These may represent such numbers as the percentage of patients for whom a drug is effective, or how long does the particular drug take to bear its effects on patients etc. These parameters are estimated using samples, as they are not known in most cases. Though the sample means themselves are good estimates of the population means, statisticians are only able to say that it could be close to the true mean with some probability. Such as, if the drug has been effective on 75% of the sampled patients, a 90% confidence interval for the true percentage may be 75% ± 3% or 72% to 78% ; implying that the true percentage of patients for whom the drug is effective has 95% probability of being in this range as suggested by the sample.
Before calculating such estimates for any population mean, one should understand what a confidence interval is. Confidence intervals display ranges of values for a specific probability that the true value of the parameter is located in within each range. These estimates are based on two facts: the probability distribution of all large sample means tend to be bell-shaped, and samples from bell-shaped distributions are more likely to be found close to their true means. So, the higher the levels of confidence, the wider these intervals.*…show more content…*

. Thanks to William Sealy Gosset (1876-1937), this theory can now be applied to estimation of means using small samples (where n < 30). He showed that small samples from bell-shaped distributions followed approximate z- distributions; and introduced the Student’s t- distributions, with adjustment for sample sizes through degrees of freedom (set as

. Thanks to William Sealy Gosset (1876-1937), this theory can now be applied to estimation of means using small samples (where n < 30). He showed that small samples from bell-shaped distributions followed approximate z- distributions; and introduced the Student’s t- distributions, with adjustment for sample sizes through degrees of freedom (set as

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