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Each sample contains different elements so the value of the sample statistic differs for each sample selected. These statistics provide different estimates of the parameter. The sampling distribution describes how these different values are distributed.

Sample size has a significant effect on sample distribution. It is often observed that small sample size results in non-normal distribution. This is a result of inadequate estimation of the dispersion of the data, and the frequency distribution does not result in a normal curve.

Increasing Sample Size With “infinite” numbers of successive random samples, the mean of the sampling distribution is equal to the population mean (µ). As the sample sizes increase, the variability of each sampling distribution decreases so that they become increasingly more leptokurtic.

As the sample size increases the t distribution becomes more and more like a standard normal distribution. In fact, when the sample size is infinite, the two distributions (t and z) are identical.

Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error.

Sample Size The larger your sample, the more sure you can be that their answers truly reflect the population. This indicates that for a given confidence level, the larger your sample size, the smaller your confidence interval.

A wide interval may earn fewer points, but it’s less risky. Wide intervals may be useful for scatterplots with weaker association.

The confidence level of the test is defined as 1 – α, and often expressed as a percentage. The width of the confidence interval decreases as the sample size increases. The width increases as the standard deviation increases. The width increases as the confidence level increases (0.5 towards 0.99999 – stronger).

The width of the confidence interval for an individual study depends to a large extent on the sample size. Larger studies tend to give more precise estimates of effects (and hence have narrower confidence intervals) than smaller studies.

In general, the narrower the confidence interval, the more information we have about the value of the population parameter. That is, the sample mean plays no role in the width of the interval. As the sample standard deviation s decreases, the width of the interval decreases.

The 95% confidence interval defines a range of values that you can be 95% certain contains the population mean. With large samples, you know that mean with much more precision than you do with a small sample, so the confidence interval is quite narrow when computed from a large sample.

The formula for calculating a z-score is is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation.

How to Find a Sample Size Given a Confidence Interval and Width (unknown population standard deviation)

1. za/2: Divide the confidence interval by two, and look that area up in the z-table: .95 / 2 = 0.475.
2. E (margin of error): Divide the given width by 2. 6% / 2.
3. : use the given percentage. 41% = 0.41.
4. : subtract. from 1.