Test of Hypothesis

You mostly know: the Standard Normal Distribution is the special case of Normal Distribution, given that: Mean: Mu=0.0 Variance: Sigma^2=1.0 Cool, as shown below: the family of normal distributions mainly vary in their mean value and/or their variance. The standard one plays the role of being the reference distribution. Well, we can convert any normal random variable to corresponding interpretations in the standard form. Hence, we simplify different computations only using standard normal distribution. OK, let's assume X is a normal distribution with mean Mu and variance Sigma^2. We can convert to a standard normally distributed random variable by following: Z=(X-Mu)/Sigma Here, we got Z as the standard normal distribution. OK, but what does this mean? Any point in X (with Mu, Sigma ) can be dealt exactly as the converted point in Z with mean Mu=0.0 and Sigma=1.0 . The numerator means: how much is the distance or differe...

Also called Gaussian distribution. OK, many things in this world tends, and should do, to be normally distributed. Any distribution is a representation of how the information or data is distributed. We mainly look for its central tendency ( mean ) and variability ( variance ). That's why the normal distribution is usually written as: N ~ (Mu, Sigma^2) For example: the weight of most adult (who still youth) people will normally be centered around some values. Yes, you right there is a diversity: some are slim and some are obese. We may expect the average weight for people (example: ages 20 to 30) to be between 70 to 74 kg. OK, let's consider it as 72 (this is the mean value). Let x represents the weight of a random person. Thus, Expected Value [x] = mean [x] = Mu = 72 kg If we have a sample, we can compute the variance (sigma^2) to indicate variability. But we may here think as following: Variance = Sigma^2 = Expected Value [(x-Mu)^2] Sta...

Variability (as name inspires) show how much values/elements are different/diverse from each others. When all values are very close to the mean value, it means we have little variability or diversity. The common used variability measure is the variance (or standard deviation , which is the positive square root of variance). Besides, the range measure can indicate the amount of variability within a group of data. I guess the formula to compute variability is known, but let's show it again: Suppose we had the following measurements data {5.2, 5.3, 4.95, 5.17, 5.22}. The diversity/variability within values seems small by inspection. Calculating the variance s^2=0.01717 which is clearly a small value. But let's see this group of data {30, 45, 39, 28, 42}. The diversity/variability within values seems large by inspection .  Calculating the variance s^2=55.7 which is clearly a large value.

Any population is usually best described by its central tendency and variability measures. Well, these parameters should be used to best interpreting for information on a specific property under study. Central tendency measures the value that mostly whole data/elements are centered around or grouped at. Which means, data values tend to be like/close to this value. The most common methodology the describe the central tendency is the mean value, also called the expected value , or average value (usually for samples). There still more methods to indicate central tendency such as mode and median . The average (or mean ) of a sample of data is just the summation divided by the data count. Whereas, the mode is the most repeated/frequent value within our data. Also the median represent the middle ordered value when sorting the data group in ascending manner. Example: the water (or juice) bottles production is a population ...

Seems obvious that larger sample size will give more close results to what happens in realty. Cool, if it's easy to select large elements to represent the sample, then we should do. Usually a sample of size 30 to 40 elements is great to have. We should increase the sample size whenever we: Can easily handle/select sample items (randomly). Perform the measurement with little effort/time consumption. Notice these are highly dependent on the problem or population nature and situations. Another point to keep is to have accurate measurements for the sample as possible. In conclusion, two important factors should be regarded: Good sample size as large as possible. This guarantees more information diversity is included in our sample. Thus it's more close to describe the whole population facts.  Using accurate measuring tools/equipment when measuring the required property of the sample because these measurements will judge next steps. This requireme...

Well, no fixed criteria in statistics. Yes, there exist many theories and methodologies to create random sequences or numbers. However, this will depend on the population nature, situations and surrounding environment. In general, we may think like following: Many persons/machines to share in sample selection is better than to be done by only one. Diversity leads to better randomness. Selecting in different times/situations/places is better than to do at once in order to get more randomness. Changing methodologies/media of selection may help. Combining two or more random samples create better random sample. Anyway, since our goal is to get accurate inferences, we should try as possible to be randomized in sample selection.

Anytime you aim to perform a study on the entire population, you will surely find that this task will be: Much time and/or efforts consuming as populations are normally huge . Impossible if the population is infinite (such as products). Here comes the role of taking samples. Yes! we just take a sample from the whole population, perform the study on the chosen sample, apply the results back to our population. This is the core of  inferential statistics because what we do is to infer parameters/properties of the population using information from a small sample. Well, this does not mean we will obtain 100% exact accurate estimations or inferences. But to be as close as possible, sample elements should be taken randomly ! At least, being random in sample selection will mostly include the diversity of information/facts within our population.