Lesson 6 Normal Distribution Flashcards
is represented by the well known bell-shaped
Normal distribution
is a distribution with population mean = 0 and standard deviation = 1.
Standard normal distribution
Essential Properties of the Normal Distribution
- Mean = Median = Mode
- It is symmetrical about the mean.
- The tails or ends are asymptotic relative
to the horizontal. - Approximately 68% of the normal population
lies within 1 standard deviation of the mean - Approximately 95 of the normal distribution lies within 2 standard deviation of the mean.
- The total area under the normal distribution curve is 1, or 100%.
- The normal distribution curve area may be subdivided into standard deviations, at least 3 units to the left and 3 units to the right of the vertical line.
- The probability that a randomly selected x of a normal population lies between two values XL and XR
z score formula
z = (x - μ) / σ
where: z = z value
x = the value of any particular observation or measurement
μ = the mean of the distribution
σ = standard deviation of the distribution
a normal distribution can be converted into standard normal distribution by obtaining the
z value
is the sign distance between a selected value, designated х, and the mean, μ, divided by the standard deviation
z value
Process of making an inference or a generalization on a population based on the results of the study on samples
Hypothesis testing
is a guess or prediction made by a researcher regarding the possible outcome of the study.
Statistical hypothesis testing
Two (2) types of statistical hypothesis
Null hypothesis (Ho)
Alternative hypothesis (Ha)
- It is the hypothesis to be tested which one hopes to reject.
- It shows equality or no significant difference or relationship between variables
- That NO statistically significant difference exists between the population parameter and the sample statistic being compared
Null hypothesis (Ho)
- It generally represents the idea which the researcher wants to prove
- Logical opposite of the null hypothesis
- That a statistically significant difference DOES EXIST between the population parameter and the sample statistic being compared
Alternative hypothesis (Ha)
Types of hypothesis testing
One-Tailed
Two-Tailed
It is a directional test with the region of rejection lying either left or right on the alternative hypothesis.
One-Tailed Test
The region of rejection is on the right tail
Right directional test
It is used when Ha uses comparatives such as
greater than,
higher than,
better than,
superior to,
exceeds, etc.
Right directional test
The region of rejection is on the left tail.
Left directional test
It is used when Ha uses comparatives such as
less than,
smaller than,
inferior to,
lower than,
below, etc.
Left directional test
It is a non-directional test with the region of rejection lying on both tails of the normal curve.
Two-Tailed Test
It is used when Ha uses word such as
Not equal to,
Significantly different, etc.
Two-Tailed Test
The error committed when the null hypothesis is rejected when in fact ** it is true** and the alternative hypothesis is false.
Type I error
The error committed when the **null hypothesis is accepted ** when in fact it is false and the alternative hypothesis is true.
Type II error
The probability of committing a Type I error is designated by
alpha (α)
the probability of committing a Type II error is designated by
beta (β).
the size of the rejection region
Alpha (α)
the size of the acceptance region.
Beta (β)
The most popular level of significance of alpha (α) are
0.01 and 0.05 levels
5 steps in testing the hypothesis
- Formulate Ho and Ha.
- Set the level of significance α, then determine the type of hypothesis test and the tabular or p-value.
- Set the criterion (when to reject Ho), determine and compute for the test statistic.
- Make your decision.
- Formulate your conclusion
Testing the hyphothesis
A. Using tabular value of Z.
- One-tailed test (right directional)
“Reject Ho if Z computed is greater than or equal to Z tabular.” - One-tailed test (left directional)
“Reject Ho if Z computed is less than or equal to Z tabular” - Two-tailed test (Z computed is positive)
“Reject Ho if Z computed is greater than or equal to Z tabular.” - Two-tailed test (Z computed is negative)
“Reject Ho if Z computed is less than or equal to Z tabular”
Testing the hyphothesis
B. Using the p-value
Reject Ho if the probability is less than or equal to alpha (α).
used when n is large, n ≥ 30
The Z-test
z = (x̄ - μ) √n / σ
Where:
σ is the population standard deviation
X is the sample mean
µ is the population mean
n is the sample size
