1. Inferential Statistics & Hypothesis Testing
Heibatollah Baghi, and
Mastee Badii

2. Objectives Conduct one sample mean test Using Z statistics
Using t statistics

3. Inferential Statistics Usage
Researchers use inferential statistics to address two broad goals:
Estimate the value of population parameters
Hypothesis testing

4. Distribution of Coin Tosses
If you see 10 heads in a row,
is it a fair coin?

5. 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).

6. 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

7. Logic of Hypothesis Testing
The further the observed value is from the mean of the expected distribution, the more significant the difference

8. What about this point?

9. What about this point?

10. Is this point part of the distribution?

11. Probability of Membership in a Distribution
Depends on location
Mean
Variance
It is a chance event

12. 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

13. 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.

14. 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.

15. 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

16. Population of IQ scores, 10-year olds
µ=100
σ=16
n = 64
Sample 1
Sample 2
Sample 3
Etc
Is sample 2 a likely
representation
of our population?

17. 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

18. 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

19. Statistical inference
Sample of
observations
Entire population of
observations
Random selection
Parameter
µ=?
Statistic
Statistical inference

20. 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

21. When Sample size is small …
A constant from
Student t Distribution
that depends on confidence
interval and sample size

22. 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.

23. TWO TYPES OF ERROR True False Reject H0 error correct decision
Fail to Reject H0

24. Alpha & Beta Errors Decision Ho is True Ho is False Reject H0 α 1-β
Fail to Reject H0
1-α β

25. 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

26. Two Types of Error Alpha: α Beta: β Probability of Type I Error
P (Rejecting Ho when Ho is true)
Beta: β
Probability of Type II Error
P (Failing to reject Ho when Ho is false)

27. Power & Confidence Level
1- β
Probability of rejecting Ho when Ho is false
Confidence level
1- α
Probability of failing to reject Ho when Ho is true

28. Steps in Test of Hypothesis
Determine the appropriate test
Establish the level of significance:α
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

29. 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

30. 2. Establish Level of Significance
α is a predetermined value
The convention
α = .05
α = .01

31. 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

32. 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)

33. 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

34. 6. Compare the Computed Test Statistic Against a Tabled Value

35. Example of Testing Statistical Hypotheses About µ When σ is Known (Large Sample Test for Population Mean).

36. “Does Home Schooling Affect Educational Outcomes?”
Research Question
“Does Home Schooling Affect Educational Outcomes?”

37. 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 σ = 50.
Home – schooled children are not required to take this test.

38. 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 > 250
α = 0.05

39. Calculated Z
Select the sample, calculate the necessary sample statistics
n=36
σ =50

40. Critical Z Determine zα  = 0.05 one sided CI of 95%
Refer to the Z table and find the corresponding Z score:
Z = 1.65

41. Make Decisions Regarding Ho
Because the calculated z is greater than the critical z, Ho is rejected.
1.80 > 1.65 and Ha is accepted
The mean of the population of home-school fourth graders is not 250.

42. Alternative Steps Step 1: Specify Hypotheses
Ho: µ = 250 Ha: µ > 250 α = .05
Step 2: Select the sample, calculate sample statistics n=36 σ =50

43. Using P value to Reject Hypothesis
Step 3: Determine the p-value . A z of corresponds to a one tailed probability of
Step 4: Make decision regarding Ho. Because the p-value of is less than α =0.05
H0 is rejected. The mean of the population of home-school fourth graders is not 250.

44. DECISION RULES In terms of z scores: If Zc > Zα Reject H0
In terms of p-value:
If p value < α Reject H0

45. 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.

46. 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.

47. Take Home Lesson
Procedures for Hypothesis Testing and Use of These Procedures in One Sample Mean Test for Normal Distribution