Sampling fundamentals
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This document discusses key concepts in sampling and statistical inference. It defines parameters and statistics, and explains sampling distributions including the sampling distribution of the mean, proportion, and difference between means. The central limit theorem is covered, stating that as sample size increases, the sampling distribution of the mean approaches a normal distribution. Common distributions used in statistical inference like the t, F, and chi-square distributions are also summarized.
Sampling fundamentals
- 1. A. Thangamani ramalingam PT, MSc(psy),PGDRM, AC spss MIAP
- 2. • Saves money ,time and energy.
- 3. • Universe/Population(finite or infinite) • sampling units • Sampling frame • Sampling design • Statistic/parameter(certain measures such as mean, median, mode or the like ones from samples, then they are called statistic(s) for they describe the characteristics of a sample. But when such measures describe the characteristics of a population, they are known as parameter(s) • Sampling error • Precision • Confidence interval
- 4. Parameter • Population characteristic e.g. μ, σ, P, median, percentiles etc Sample statistic • Any quantity computed from values in a sample e.g. x, s, sample proportion etc.
- 5. • Concept of a sampling distribution is perhaps the most basic concept in inferential statistics. • The sampling distribution of the mean • The sampling distribution of the difference between means • The sampling distribution of r • The sampling Distribution of a Proportion • Central Limit Theorem The central limit theorem states that: • Given a population with a finite mean μ and a finite non zero variance σ2, the sampling distribution of the mean approaches a normal distribution with a mean of μ and a variance of σ2/N as N, the sample size, increases
- 6. • Example of the Central Limit Theorem in Practice: Roll 30 dice and calculate the average (sample mean) of the numbers that you get on each die. Now repeat this experiment 1000 times each time rolling 30 dice and computing a new sample mean. Plot a histogram of the 1000 sample means that you have obtained. This plot will look approximately normal
- 8. Sampling distribution of mean refers to the probability distribution of all the possible means of random samples of a given size that we take from a population. If samples are taken from a normal population
- 9. • Usually the statistics of attributes correspond to the conditions of a binomial distribution that tends to become normal distribution as n becomes larger and larger. If p represents the proportion of defectives i.e., of successes and q the proportion of non defectives i.e., of failures (or q = 1 – p) and if p is treated as a random variable, then the sampling distribution of proportion of successes has a mean = p with standard deviation = p × q n, where n is the sample size.
- 10. • The variable t differs from z in the sense that we use sample standard deviation in the calculation of t, whereas we use standard deviation of population in the calculation of z. There is a different t distribution for every possible sample size i.e., for different degrees of freedom. The degrees of freedom for a sample of size n is n – 1. As the sample size gets larger, the shape of the t distribution becomes approximately equal to the normal distribution
- 11. • Two independent normal populations, having the same variance • The calculated value of F from the sample data is compared with the corresponding table value of F and if the former is equal to or exceeds the latter, then we infer that the null hypothesis of the variances being equal cannot be accepted. We shall make use of the F ratio in the context of hypothesis testing and also in the context of ANOVA technique.
- 12. • Distribution is not symmetrical and all the values are positive with (n – 1) degrees of freedom. • Chi-square distribution is encountered when we deal with collections of values that involve adding up squares.
- 13. • A point estimator is a formula that uses sample data to calculate a single number (a sample statistic) that can be used as an estimate of a population parameter. e.g. ¯x, s to calculate μ, σ,