Slide 1 Testing Statistical HypothesisThe One Sample t-Test

Heibatollah Baghi, and

Mastee Badii

Slide 2 ### Parametric and Nonparametric Tests

Slide 3 ### Parametric and Nonparametric Tests

- Nonparametric tests are easier to compute and have less restrictive assumptions
- Parametric tests are much more powerful (less likely to have type II error)

What is type two error?

This lecture focuses on

One sample t-test

which is a parametric test

Slide 4 ### Two Types of Error

- Alpha: α
- Probability of Type I Error
- P (Rejecting Ho when Ho is true)
- Predetermined Level of significance

- Beta: β
- Probability of Type II Error
- P (Failing to reject Ho when Ho is false)

Slide 5 True

False

True

(Accept Ho)

Type II error

Probability = b

False

(Rejects Ho)

Type I error

Probability = a

### Types of Error in Hypothesis Testing Ho: Hand-washing has no effect on bacteria counts

Slide 6 True

False

True

(Accept Ho)

Correct decision

Probability = 1- a

Type II error

Probability = b

False

(Rejects Ho)

Type I error

Probability = a

Correct decision

Probability = 1- b

### Types of Error in Hypothesis Testing Ho: Hand-washing has no effect on bacteria counts

Slide 7 True

False

True

(Accept Ho)

Correct decision

Probability = 1- a

Type II error

Probability = b

False

(Rejects Ho)

Type I error

Probability = a

Correct decision

Probability = 1- b

### Power & Confidence Level

- Power
- 1- β
- Probability of rejecting Ho when Ho is false

- Confidence level
- 1- α
- Probability of failing to reject Ho when Ho is true

Slide 8 ### Level of Significance

- α is a predetermined value by convention usually 0.05
- α = 0.05 corresponds to the 95% confidence level
- We are accepting the risk that out of 100 samples, we would reject a true null hypothesis five times

Slide 9 Population of IQ scores, 10-year olds

µ=100

σ=16

n = 64

Sample 1

Sample 2

Sample 3

Etc

### Sampling Distribution Of Means

Slide 10 ### Sampling Distribution Of Means

- A sampling distribution of means is the relative frequency distribution of the means of all possible samples of size n that could be selected from the population.

Slide 11 ### One Sample Test

- Compares mean of a sample to known population mean

This lecture focuses on

one sample t-test

Slide 12 The One Sample t – Test

Testing statistical hypothesis about µ when σ is not known OR sample size is small

Slide 13 ### An Example Problem

- Suppose that Dr. Tate learns from a national survey that the average undergraduate student in the United States spends 6.75 hours each week on the Internet – composing and reading e-mail, exploring the Web and constructing home pages. Dr. Tate is interested in knowing how Internet use among students at George Mason University compares with this national average.
- Dr. Tate randomly selects a sample of only 10 students. Each student is asked to report the number of hours he or she spends on the Internet in a typical week during the academic year.

Population mean

Small sample

Population variance is unknown & estimated from sample

Slide 14 ### 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

Slide 15 ### 1. Determine the appropriate test

- If sample size is more than 30 use z-test
- If sample size is less than 30 use t-test

Slide 16 ### 2. Establish Level of Significance

- α is a predetermined value
- The convention

In this example, assume α = 0.05 Slide 17 ### 3. Determine Whether to Use a One or Two Tailed Test

- H0 :µ = 6.75
- Ha :µ ≠ 6.75

A two tailed

test because it

can be either larger

or smaller

Slide 18 ### 4. Calculating Test Statistics

Sample mean

Slide 19 ### 4. Calculating Test Statistics

Deviation

from sample

mean

Slide 20 ### 4. Calculating Test Statistics

Squared

deviation

from

sample

mean

Slide 21 ### 4. Calculating Test Statistics

Standard deviation of observations

Slide 22 ### 4. Calculating Test Statistics

Calculated

t value

Slide 23 ### 4. Calculating Test Statistics

Standard deviation

of sample means

Slide 24 ### 4. Calculating Test Statistics

Calculated t

Slide 25 ### 5. Determine Degrees of Freedom

- Degrees of freedom, df, is value indicating the number of independent pieces of information a sample can provide for purposes of statistical inference.
- Df = Sample size – Number of parameters estimated
- Df is n-1 for one sample test of mean because the population variance is estimated from the sample

Slide 26 ### Degrees of Freedom

- Suppose you have a sample of three observations:

X

--------

--------

--------

Slide 27 ### Degrees of Freedom

- Why n-1 and not n?
- Are these three deviations independent of one another?
- No, if you know that two of the deviation scores are -1 and -1, the third deviation score gives you no new independent information ---it has to be +2 for all three to sum to 0.

Slide 28 ### Degrees of Freedom Continued

- For your sample scores, you have only two independent pieces of information, or degrees of freedom, on which to base your estimates of S and

Slide 29 ### 6. Compare the Computed Test Statistic Against a Tabled Value

- α = .05
- Df = n-1 = 9
- Therefore, reject H0

Slide 30 Decision Rule for t-Scores

If |tc| > |tα| Reject H0

Slide 31 Decision Rule for P-values

If p value < α Reject H0

Pvalue is one minus

probability of observing

the t-value calculated

from our sample

Slide 32 ### Example of Decision Rules

Slide 33 ### Constructing a Confidence Interval for µ

Standard deviation of sample means

Sample mean

Critical t value

Slide 34 ### Constructing a Confidence Interval for µ for the Example

- Sample mean is 9.90
- Critical t value is 2.262
- Standard deviation of sample means is 1.29
- 9.90 + 2.262 * 1.29
- The estimated interval goes from 6.98 to 12.84

Slide 35 ### Distribution of Mean of Samples

In drawing samples at random, the probability is .95 that an interval constructed with the rule

will include m

Slide 36 ### Sample Report of One Sample t-test in Literature

One Sample t-test Testing Neutrality of Attitudes Towards Infertility Alternatives

Slide 37 N

Mean

Std. Deviation

Std. Error Mean

Number of Hours

10

9.90

4.067

1.286

Test Value = 6.75

tc

df

Sig. (2-tailed)

Mean Difference

95% Confidence Interval of the Difference

Lower

Upper

Number of Hours

2.449

9

.037

3.150

.24

6.06

### Testing Statistical Hypothesis With SPSS

SPSS Output: One-Sample Statistics

One-Sample Test

Slide 38 Take Home Lesson

Procedures for Conducting & Interpreting One Sample Mean Test with Unknown Variance