A normal distribution can be used to describe a variety of variables, such as heights, weights, and temperatures. A normal distribution is bell-shaped, unimodal, symmetric, and continuous; its mean, median, and mode are equal. Since each variable has its own distribution with mean m and standard deviation s, mathematicians use the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Other approximately normally distributed variables can be transformed to the standard normal distribution with the formula z =(X -μ)/σ.

Gaussian Function or Bell Curve is the graph of the associated probability density function is bell shape which has the peak at the mean. The curve was developed in 1733 by Abraham de Moivre, which was used as an approximation to the binomial distribution.

When the data values are evenly distributed about the mean, a distribution is said to be a symmetric distribution.

When the majority of the data values fall to the right of the mean, the distribution is said to be a negatively or left-skewed distribution. The mean is to the left of the median, and the mean and the median are to the left of the mode.

When the majority of the data values fall to the left of the mean, a distribution is said to be a positively or right-skewed distribution. The mean falls to the right of the median, and both the mean and the median fall to the right of the mode.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The z value is actually the number of standard deviations that a particular X value is away from the mean. All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score as shown in the left figure.

A normal distribution is often used to solve problems that involve the binomial distribution since when n is large (say, 100), the calculations are too difficult to do by hand using the binomial distribution.

Also, recall that a binomial distribution is determined by n (the number of trials) and p (the probability of a success). When p is approximately 0.5, and as n increases, the shape of the binomial distribution becomes similar to that of a normal distribution. The larger n is and the closer p is to 0.5, the more similar the shape of the binomial distribution is to that of a normal distribution.

A correction for continuity is a correction employed when a continuous distribution is used to approximate a discrete distribution

Hypothesis Testing is a statistical method that is used in making statistical decisions using experimental data. These include ssumptions made about population parameter.

Statistical hypothesis is a conjecture about a population. There are two types of statistical hypotheses: the null and the alternative hypotheses. The null hypothesis states that there is no difference, and the alternative hypothesis specifies a difference.

The three methods used to test hypotheses are a.) The traditional method b.) The P-value method and c.) The confidence interval method.

A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. The numerical value obtained from a statistical test is called the test value.

In the hypothesis-testing situation, there are four possible outcomes. In reality, the null hypothesis may or may not be true, and a decision is made to reject or not reject it on the basis of the data obtained from a sample. The four possible outcomes are shown in the right figure.

Note that a type I error occurs if you reject the null hypothesis when it is true. On the other hand, a type II error occurs if you do not reject the null hypothesis when it is false.

The level of significance is the maximum probability of committing a type I error. This probability is symbolized by α (Greek letter alpha). That is, P(type I error) = α

The critical value separates the critical region from the noncritical region. The symbol for critical value is C.V. The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected. The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.

A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. A one-tailed test is either a righttailed test or left-tailed test, depending on the direction of the inequality of the alternative hypothesis.

In a two-tailed test, the null hypothesis should be rejected when the test value is in either of the two critical regions.

Shown in the left figure are the critical and non critical regions for a one-tailed and two-tailed test.

In a simple relationship, there are two variables—an independent variable, also called an explanatory variable or a predictor variable, and a dependent variable, also called a response variable. A simple relationship analysis is called simple regression, and there is one independent variable that is used to predict the dependent variable.

In a multiple relationship, called multiple regression, two or more independent variables are used to predict one dependent variable.

Simple relationships can also be positive or negative. A positive relationship exists when both variables increase or decrease at the same time. For instance, a person’s height and weight are related; and the relationship is positive, since the taller a person is, generally, the more the person weighs. In a negative relationship, as one variable increases, the other variable decreases, and vice versa.

The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables. The symbol for the sample correlation coefficient is r. The symbol for the population correlation coefficient is r (Greek letter rho).

Shown in the right figure is the formula used to obtain correlation coefficient.

If the value of the correlation coefficient is significant, the next step is to determine the equation of the regression line, which is the data’s line of best fit.

Shown in the left figure is the formula used to compute the equation for regression line equation.