Stat 4 the normal distribution & steps of testing hypothesis

The Normal Distribution

Also called Gaussean Distribution.Mean = Median = Mode.Skewness = zero & Kurtosis =

zero.Total area under the curve = 1.

The Normal Distribution

The Normal Distribution, cont.

68% of observations lie between minus & plus one SD. ( -1Z & +1Z ).

95% of observations lie between minus 1.96 & plus 1.96 SD units.

99% of observations lie between minus 2.58 & plus 2.58 SD units.

99.7% of observations lie between minus 3 & plus 3 SD units.

Confidence interval for a mean

The range with in which the population mean is likely to lie.

s.d. 95% C.I. = X ± 1.96

√n s.d.

99% C.I. = X ± 2.58

√n

Confidence interval for a mean

If n ≤ 30

s.d. C.I. = X ± t

√n

Steps of testing the statistical hypothesis

*Assumption We assume that our population(s)

are normally distributed.*Hypothesis We put null hypothesis (Ho) &alternative hypothesis (HA).

*Levels of significance (alpha)Alpha = the probability of rejecting a

true null hypothesis. Usually alpha = 0.05 or 0.01*Degrees of freedom (d.f.): Depends on

the type of the statistical test.*The statisticsDepends on type of data.

*Statistical decisionWhether to reject or not to reject Ho.

*P valueWhether < or > 0.05 (or whether P < or > 0.01)

Tests of Statistical Significance

Depending on the type of data, an

appropriate test will be used.Generally speaking, Data are

either numerical or categorical data.

• For Numerical data; we usually compute the mean & it’s standard deviation.

• In order to test whether there is a significant difference between two means related to two different groups, we use the student (t) test.

The t test is also used to compare between two sets of data within the same group ie. To compare between two readings for the same person but on two occasions, eg. Before & after treatment.

*To compare between several means; we use the analysis of variance (ANOVA, or called the F test); & when we have several numerical variables & one of them is dependent on the others (independent variables) then we use the multiple regression.

*In order to examine the nature & strength of the relationship between two variables ( e.g.. Blood pressure & age), simple linear regression & correlation tests are used.

*The objective of regression analysis is to predict (estimate) the value of one variable corresponding to a given value of another variable.

*Correlation analysis is concerned with measuring the strength of the relationship between variables.