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Inferential Statistics Hypothesis Testing

- Heibatollah Baghi, and
- Mastee Badii

Objectives

- Conduct one sample mean test
- Using Z statistics
- Using t statistics

Inferential Statistics Usage

- Researchers use inferential statistics to address

two broad goals - Estimate the value of population parameters
- Hypothesis testing

Distribution of Coin Tosses

If you see 10 heads in a row, is it a fair coin?

Sample Population

- Think of any sequence of throws as a sample from

all possible throws - Think of all possible throws as the entire

population. - One-Sample Inferential Tests estimate the

probability that a sample is representative of

the total population (within /- 2 standard

deviations of the mean, or the middle 95 of the

distribution).

Logic of Hypothesis Testing

- Is the value observed consistent with the

expected distribution? - On average, 100 coin tosses should lead to 50/50

chance of heads. - Some coin tosses will be outliers, giving

significantly different results. - Are differences significant or merely random

variations?

Statistics is the art of making sense of

distributions

Logic of Hypothesis Testing

- The further the observed value is from the mean

of the expected distribution, the more

significant the difference

What about this point?

What about this point?

Is this point part of the distribution?

Probability of Membership in a Distribution

- Depends on location
- Mean
- Variance
- It is a chance event

One-Sample Tests

- We set a standard beyond which results would be

rare (outside the expected sampling error) - We observe a sample and infer information about

the population - If the observation is outside the standard, we

reject the hypothesis that the sample is

representative of the population

Random Sampling

- A simple random sampling procedure is one in

which every possible sample of n objects is

equally likely to be chosen. - The principle of randomness in the selection of

the sample members provides some protection

against the sample unrepresentative of the

population. - If the population were repeatedly sampled in this

fashion, no particular subgroup would be over

represented in the sample.

Sampling Distribution

- The concept of a sampling distribution, allows us

to determine the probability that the particular

sample obtained will be unrepresentative. - On the basis of sample information, we can make

inference about the parent population.

Sampling Distribution

- Sampling Error.
- No sample will have the exact same mean and

standard deviation as the population - Sampling distribution of the mean
- In research sampling error is often unknown since

we do not have the population parameters - A distribution of means of several different

samples of our population - Less widely distributed than the population
- Usually Normal

Population of IQ scores, 10-year olds

µ100 s16

n 64

Sample 2

Sample1

Sample 3

Etc

Is sample 2 a likely representation of our

population?

Distribution of Sample Means

- The mean of a sampling distribution is identical

to mean of raw scores in the population (µ) - If the population is Normal, the distribution of

sample means is also Normal - If the population is not Normal, the distribution

of sample means approaches Normal distribution as

the size of sample on which it is based gets

larger

Central Limit Theorem

Standard Error of the Mean

- The standard deviation of means in a sampling

distribution is known as the standard error of

the mean. - It can be calculated from the standard deviation

of observations - The larger our sample size, the smaller our

standard error

Sample of observations

Entire population of observations

Random selection

Parameter µ?

Statistic

Statistical inference

Estimation Procedures

- Point estimates
- For example mean of a sample of 25 patients
- No information regarding probability of accuracy
- Interval estimates
- Estimate a range of values that is likely
- Confidence interval between two limit values
- The degree of confidence depends on the

probability of including the population mean?

When Sample size is small

A constant from Student t Distribution that

depends on confidence interval and sample size

HYPOTHESIS TESTING

- Hygiene procedures are effective in preventing

cold. - State 2 hypotheses
- Null H0 Hand-washing has no effect on bacteria

counts. - Alternative Ha Hand-washing reduces bacteria.
- The null hypothesis is assumed true i.e., the

defendant is assumed to be innocent.

TWO TYPES OF ERROR

Alpha Beta Errors

Two Types of Error in Admission to ICU

- Correct decisions
- Patients admitted to ICU who would have failed if

otherwise - Patients denied admission who do fine in step

down unit - Errors
- Patient admitted who does not need to be there
- Patient denied admission who needs to be there

Two Types of Error

- Alpha a
- Probability of Type I Error
- P (Rejecting Ho when Ho is true)
- Beta ß
- Probability of Type II Error
- P (Failing to reject Ho when Ho is false)

Power Confidence Level

- Power
- 1- ß
- Probability of rejecting Ho when Ho is false
- Confidence level
- 1- a
- Probability of failing to reject Ho when Ho is

true

Steps in Test of Hypothesis

- Determine the appropriate test
- Establish the level of significancea
- Determine whether to use a one tail or two tail

test - Calculate the test statistic
- Determine the degree of freedom
- Compare computed test statistic against a tabled

value

1. Determine Appropriate Test

- Level of measurement
- Number of groups being compared
- Sample size
- Extent to which assumption for parametric tests

have been met - Relatively Normal distribution
- Approximately interval level variable

2. Establish Level of Significance

- a is a predetermined value
- The convention
- a .05
- a .01
- a .01

3. Determine Whether to Use One or Two Tailed Test

- If the alternative hypothesis specifies direction

of the test, then one tailed - Otherwise, two tailed
- Most cases

4. Calculating Test Statistics

- For one sample tests, use Z test statistic if

population is Normal, ? is known, or if sample

size is large - For one sample tests, use T static if population

distribution is not known or if sample size is

small (less than 30)

5. Determine Degrees of Freedom

- Number of components that are free to vary about

a parameter - Df Sample size Number of parameters estimated
- Df is n-1 for one sample test of mean

6. Compare the Computed Test Statistic Against a

Tabled Value

Example of Testing Statistical Hypotheses About µ

When s is Known(Large Sample Test for Population

Mean).

Research Question

- Does Home Schooling Affect Educational Outcomes?

Statistical Hypotheses

- Dr. Tate, a researcher at GMU decided to conduct

a study to explore this question. He found out

that every fourth-grade student attending school

in Virginia takes CAT. - Scores of CAT are normally distributed with µ

250 and s 50. - Home schooled children are not required to take

this test.

Statistical Hypotheses

- Dr. Tate selects a random sample of 36 home

schooled fourth graders and has each child

complete the test. (It would be too expensive and

time-consuming to test the entire population of

home-schooled fourth-grade students in the sate.) - Step 1 Specify Hypotheses
- H0 µ 250
- Ha µ gt 250
- a 0.05

Calculated Z

- Select the sample, calculate the necessary sample

statistics - n36
- s 50

Critical Z

- Determine za
- ? 0.05 one sided
- CI of 95
- Refer to the Z table and find the corresponding Z

score - Z 1.65

Make Decisions Regarding Ho

- Because the calculated z is greater than the

critical z, Ho is rejected. - 1.80 gt 1.65 and Ha is accepted
- The mean of the population of home-school fourth

graders is not 250.

Alternative Steps

- Step 1 Specify Hypotheses
- Ho µ 250 Ha µ gt 250 a .05
- Step 2 Select the sample, calculate sample

statistics n36 s 50

Using P value to Reject Hypothesis

- Step 3 Determine the p-value . A z of 1.80

corresponds to a one tailed probability of

0.036. - Step 4 Make decision regarding Ho. Because the

p-value of 0.036 is less than a 0.05 - H0 is rejected. The mean of the population of

home-school fourth graders is not 250.

DECISION RULES

- In terms of z scores
- If Zc gt Za Reject H0
- In terms of p-value
- If p value lt a Reject H0

The One-sample Z Test

- One-Sample tests of significance are used to

compare a sample mean to a (hypothesized)

population mean and determine how likely it is

that the sample came from that population. We

will determine the extent to which they occur by

chance. - We will compare the probability associated with

our statistical results (i.e. probability of

chance) with a predetermined alpha level.

The One-sample Z Test

- If the probability is equal to or less than our

alpha level, we will reject the null hypothesis

and conclude that the difference is not due to

chance. - If the probability of chance is greater than our

alpha level, we will retain the null hypothesis

and conclude that difference is due to chance.

Take Home Lesson

- Procedures for Hypothesis Testing and Use of

These Procedures in One Sample Mean Test for

Normal Distribution

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