What is the difference between Large and small sample?

Answer 1

0.7.1 Difference between Small and Large Samples:-

Though it is difficult t draw a clear-cut line of demarcation between large and small samples it is normally agreed amongst statisticians that a sample is to be recorded as large only if its size exceeds 30. The tests of significance used for dealing with problems samples for the reason that the assumptions that we make in case of large samples do not hold good for small samples.

The assumption made while dealing with problems relating to large samples are:-

(i) The random sampling distribution of a statistic is approximately normal. and

(ii) Values given by the samples are sufficiently close to the population value and can be used in its place for calculating the standard error of the estimate.

Fourmula

0.7.2 (Large Sample) Testing the significance of the difference between the means of two samples.)

To compare the means of two population we must understand the theory concerning the distribution of differences of sample means. Statisticians have determined that the distribution distribution difference between mean d (d Mean's) is approximately normal for large samples of n1 and n2. That is the distribution of differences of sample means is normal as long as neither n1 nor n2 Is less than 30. We can therefore use the probabilities associated with the normal distribution to construct confidence intervals and to perform hypothesis tests associated with this distribution.

PROCUEDURS:-

1. To compare the (μ1) mean of population 1 with the mean (μ2), of population 2 two independent random random samples of sizes n1 and n2 are to be selected from population 1 and population 2 respectively.

By independent we mean that the sample drawn from population 1, in no way affects the sample drawn from population 2 fro example drawing two samples from men population and women population

2. Compute (Mean1) and (Mean 2) i.e., mean of the sample 1 and 2

3. Computer the difference in the two samples means, d (mean) i.e,. d(Mean) = (Mean1 -Mean2).

Thus for each pair of sample means of (Mean1) and (Mean2). a value of d(Mean) is obtained. The result is therefore a distribution of d(Mean)s.

4. If μ1 and σ1 are the parameters of population 1. and μ2 and σ2 are the parameters of population 2, then for the distribution of d(Mean)s the menu μd(Mean)s is given by the equation

μd(Mean)s = μ1 - μ2 the mean of the difference of the distribution of mean is the difference of the means of the two populations being compared.

5. The standard deviation (or standard error) of the distribution of d(mean)s (written as σd(Mean)s) is given by the equation

(Large Sample) Testing the significance of the difference between the means of two samples.)

1. Point Estimation:- According to Central Limit Theorem for large samples the means of sampling distribution are normally distributed. The procedure that is frequently used to obtain a point estimate for the m of some population involves the following steps:

(a) Select a representation (random) sample of the population.

(b) Determine the mean (Mean) of the sample data

(c) Assert that the value of M is the corresponding value of (Mean) i.e., = μ.

2. Interval Estimation:-

An extension of the above method of obtaining an estimate for μ is with the confidence interval, i.e., an interval estimate for μ.

The advantages of interval estimate are:

1. Interval estimate is more likely to be correct than the point estimate.

2. We can calculate the probability that a given interval contains the mean of a population. We therefore speak of a specific interval as having "90' per cent probability of containing μ.

3. We can choose the value of the probability we want for a given interval before we actually construct it.

Recall that the central limit theorem asserts that for large sample sizes, the means are normally distributed. Furthermore, we know that any given mean (Mean) value can be standardized with the equation.

Where μ = Mean of the population

μ.(Mean) = Mean of the sampling distribution of means.

σ (Mean) = standard error or sampling distribution

Since

μ.(Mean) μ we can write the following equation

Now, with a given pair of Z values associated with some percentage of the Z distribution and equation, we can determine an upper and lower boundary for the same percentage of (Mean) values in the given mean distribution.