Normal Distribution Is Very Much

Normal distribution is very much what it sounds like. This distribution is symmetrical and is shaped like a bell when graphed on the Cartesian plane. The normal distribution has the mean figure, the median figure and the mode all basically located at the same place on the distribution. This occurs at the peak and the frequencies will gradually decrease at both ends of this bell shaped curved.

Unfortunately this is simply a model of looking at a problem and no definite predictions can be made with this or any other statistical tool, however this model does have real practical value. Many things in life follow this model and are normally distributed offering a least a guide in how to best understand and predict behavior mathematically using statistics. Suppose X is normal with mean ? And variance ?2. Any probability involving X can be computed by converting to the z-score, where Z = (X? )/?

Eg: If the mean IQ score for all test-takers is 100 and the standard deviation is 10, what is the z-score of someone with a raw IQ score of 127?the z-score defined above measures how many standard deviations X is from its mean. The z-score is the most appropriate way to express distances from the mean. For example, being 27 points above the mean is useful if the standard deviation is 10, but not so great if the standard deviation is 20. (z= 2.7, vs. z= 1.35).

Question 2 The central limit theorem states that the distribution of the sum of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution. The importance of the central limit theorem is very widespread as it is the reason that many statistical procedures work. Regardless of the population distribution model, as the sample size increases, the sample mean tends to be normally distributed around the population mean, and its standard deviation shrinks as n increases.

To use the central limit theorem the sample size must be independent and large enough so a decent amount of data can be formulated to utilize this statistical tool. When taking samples using the central limit theorem each one should represent a random sample from the population or follow the population distribution. The samples size should also be less than ten percent of the entire population. Simple random sampling refers to any sampling method that consists of a population with N.

objects, the sample consists of n objects and if all the possible samples of n objects are equally likely to occur, the sampling method is called simple random sampling. This method allows researchers to use methods to analyze sample results. Confidence intervals are created that deviate from the sample mean to help model their situation. For example if there is a population with probability p of a certain characteristic (and q=1-p). We have a random sample of n from the population.

We can find the mean and standard deviation of the proportion of that sample that has the characteristic. X 1, X2, ..., Xn are n random variables that are independent and identically distributed with mean ? And standard deviation ? Sn= X1+X2+...+Xn is the sample sum. We can show E (Sn)=n? And SD (Sn)=? n. CLT states: Sn-n? /0,1: as n Question 3 Point estimates summarize the sample by a single number that is an estimate of the population parameter.

An interval estimate is a range of values within which the true parameter lies with higher probability. In any estimation problem, we need to obtain both a point estimate and an interval estimate. The point estimate is our best guess of the true value of the parameter, while the interval estimate gives a measure of accuracy of that point estimate by providing an interval that contains plausible values.

To construct a confidence interval for a single unknown population mean ?, where the population standard deviation is known, we need x-as an estimate for ? And we need the margin of error. The margin of error is called the error bound for a population mean (EBM). The sample mean x? is the point estimate of the unknown population mean ?

The confidence interval estimate will have the form:(point estimate - error bound, point estimate + error bound) or, in symbols,(x? EBM, x?+EBM) the margin of error depends on the confidence level (CL). The confidence level is often considered the probability.