Normal Distribution and Statistical Hypothesis Testing Flashcards
is a distribution that is symmetric about the mean, continuous variables
Normal Distribution/ Probability Distribution Curve
mean, median, mode are approximately same
Probability distribution curve
Normal (no skewed)/ Gaussian Curve
(Graphical distribution of probability
Data is spread normally)
mean, median, mode are not same
Many outlier that are lower than mean
Negatively skewed
(Mean < median < mode (mean and median is greater than the mean)
Left tail is dragging mean)
Many outliers that are higher than the mean
Mode < median < mean
Positively skewed
Data set that has extreme values larger than the mean
Right tail is dragging mean
distortion
Asymmetry that deviates from the symmetrical bell-curve/ normal distribution
Imbalance in the distribution of data relative to the mean
Skewness
Values plotted will be -3 to +3
extreme observations/ values, far from the values observed
Outliers
measures the peak or flatness of data set
Kurtosis
negative kurtosis, flat/ low peak, short tail
Platykurtic
normal distribution (kurtosis)
Mesokurtic
Characteristics of a Normal Distribution:
follows a consistent pattern
Earliest distribution to be well studied
First equation is derived by Abraham de Moivre
Bell shaped and symmetric about the mean
(Friedrich Gauss: work became well known
Symmetrical and mesokurtic)
Characteristics of a Normal Distribution:
Mean = median = mode
____: center of the curve
____: only 1, peak of curve
Mean
Mode
Characteristics of a Normal Distribution:
Total area under the curve (AUC) is ____%
1 or 100
Characteristics of a Normal Distribution:
Has long tapering tails that extend infinitely in either direction but never touching the x-axis
____: lines that as it gets closer but never reach/ touch the x-axis
Asymptote
Characteristics of a Normal Distribution:
Completely determined by two parameters, its ___ (μ) and __ (𝜎, sigma)
____: location of the curve in x-axis
____: spread of curve
Mean
SD
(𝜎 increases, distribution becomes wider; low peak
𝜎 decreases, distribution becomes thinner; high peak)
1 SD covers __% of the distribution
2 SD covers __% of the distribution
3 SD covers __% of the distribution
68
95
99.7
Importance of Normal Distribution:
Useful for explaining many ______ _______
Even if the distribution of the variable is not normal, can easily transform using log, square root or other transformation to make it approximate the normal distribution
Most measurements/ variables are normally distributed
biological phenomena
Importance of Normal Distribution:
Plays an important role in ______ ______because:
Many statistical test assume normality of the distribution
The other important distributions (binomial, t-distribution) can be approximated by the normal esp. when the sample size is large enough
The sampling distribution of the mean is approximately normal if the sample size is large enough (central limit theorem)
statistical inference
Mean =0, SD =1
Capital ‘Z’ is traditionally used to represent the standard normal random variable
Small letter ‘z’ is used to represent a particular value of Z
The Standard Normal Distribution
Areas under the standard normal are tabulated
Any value x from the normal distribution can be transformed into a standard normal value of z using the formula
Statement about the value of a population parameter
Mean, median, mode, variance, SD, proportion, total
Statistical Hypothesis
Assertion or proposition about the relationship between 2 or more variables
Concerned w/ the parameters of population
Statistical Hypothesis
Formulated as a result of years of observation and research
Method of making decision using data, whether from a controlled experiment or an observational study (not controlled)
Statistical Hypothesis
Set of procedures that culminates in either rejection or non-rejection of null hypothesis
Involves the comparison of two hypotheses: Null and Alternative
Statistical Hypothesis
Hypothesis: prediction, educated guess of the results, should be testable
p< alpha: probability of occurrence of sample results is low, _____ null hypothesis
p> alpha: probability of occurrence of sample results is high, _____ null hypothesis
reject
not reject
7 Steps of Hypothesis Testing
SSSDCMD
State the null (H0) & alternative (H1 or H0) hypothesis
State the Level of Significance
Select the appropriate test statistic/ test criterion
Determine the Critical Region or Region of Rejection
Compute the test-statistic
Make a Statistical Decision
Draw conclusion
(H0): Statement of equality
Hypothesis of no difference
Null hypothesis
Often used to signify zero treatment effect or the equivalence of population parameters or to a specified value
Usually the research hypothesis (the hypothesis the investigator believes in)
What you believe to be true in population
Alternative hypothesis (Ha)
(Uses or ≠ sign
If P is low, then the null (H0) must go)
Ho vs Ha:
The proportion of students who obtain a grade of 2.0 or better among those using models for demonstration, PM, is equal to the proportion among those using cadavers, PC
Ho: PM = PC (”no difference”, equal)
Ho vs Ha:
The proportion of students who obtain a grade of 2.0 or better among those using models for demonstration, PM , is not equal to the proportion among those using cadavers, PC
Ha: PM ≠ PC (”two-tailed test”, not equal)
Ho vs Ha:
The proportion of students who obtain a grade of 2.0 or better among those using models for demonstration, PM , is greater than to the proportion among those using cadavers, PC
HA: PM > PC (”one-tailed test”, greater or less than; can also use
Types of Ha:
Simply states that there is difference in the groups being compared
But did not indicate w/c one is greater or less than
Non-directional (two-tailed test, not equal):
Types of Ha:
Specifies direction if disagreement with H0
Has a direction, says which one is longer or shorter
Directional (one-tailed test)
(𝛂); alpha (set before data collection, low value)
Is set arbitrarily by research before collecting data
Level of significance
Usually set as 0.05, 0.1 or 0.01
Types of Error:
Error of rejecting a true null hypothesis
Alpha is kept at a low level when it is important not to make a mistake of rejecting a true H0
Higher alpha, lower beta
Type I error (𝛼)
Types of Error:
Error of not rejecting a false null hypothesis
Beta is kept at a low level when it is important not to accept a false H0
Probability of not rejecting a false null hypothesis
Inversely related to alpha: lower alpha, higher beta
Type II error (𝜷)
Found at the tail end of distribution
Set of value of test statistics which leads to the rejection of the null hypothesis
Value of test statistics that corresponds to alpha (level of significance)
Area of Rejection
Basic formula of the test statistic: numerical data from the sample:
Formula: test statistic = observed statistic - expected parameter under H0/ standard error
always towards the null hypothesis
Whether to reject or not reject the null hypothesis
Statistical decisions
Ways for Making Statistical Decision:
Compare _____ with the_____
Where did the test statistic fall: critical or non-rejection region
test statistic , critical region
Ways for Making Statistical Decision:
Compare _____ with the_____
Statistical software automatically give this value
p-value , a(level of significance)
Rejection or Non-rejection of Ho:
“There is sufficient evidence to say that (alternative hypothesis)”
Ex: There is sufficient evidence to say that the proportion of students who obtain a grade of 2.0 or better among those using models for demonstration, PM, is not equal to the proportion among those using cadavers, PC
Rejection of the Null Hypothesis (H0)
Rejection or Non-rejection of Ho:
“There is no sufficient evidence to say that (alternative hypothesis)”
Ex: There is no sufficient evidence to say that the proportion of students who obtain a grade of 2.0 or better among those using models for demonstration, PM, is not equal to the proportion among those using cadavers, PC
Non-rejection of the Null Hypothesis (H0) leads to a conclusion stated this way
Reject or Do not reject or Non-rejection: (Ho)
Sample does not come from a population with the same parameter values defined by null hypothesis
Sample value cannot support/ not consistent with the null hypothesis
Reject H0
Reject or Do not reject or Non-rejection: (Ho)
Conclude that there is no sufficient evidence to say that Ha is NOT true
H0 is not accepted
Do not reject H0
Reject or Do not reject or Non-rejection: (Ho)
Not proof that the null hypothesis is correct
Non-rejection H0
Factors that result to non-rejection:
Insufficient ____ to conclude Ha rather than proof of H0
Inadequate ____
____ problems
proof
sample size
measurement
