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### The One Sample t – Test

### Decision Rule for t-Scores

### Decision Rule for P-values

### Take Home Lesson

Parametric and Nonparametric Tests

- Parametric tests estimate at least one parameter (in t-test it is population mean)

Usually for normal distributions and when the dependent variable is interval/ratio

- Nonparametric tests do not test hypothesis about specific population parameters

Distribution-free tests

Although appropriate for all levels of measurement most frequently applied for nominal or ordinal measures

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

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)

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 countsFalse

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 countsFalse

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

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

Population of IQ scores, 10-year olds

µ=100

σ=16

n = 64

Sample 1

Sample 2

Sample 3

Etc

Sampling Distribution Of MeansSampling 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.

One Sample Test

- Compares mean of a sample to known population mean
- Z-test
- T-test

This lecture focuses on

one sample t-test

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

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

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

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
- Sample size of 10

2. Establish Level of Significance

- α is a predetermined value
- The convention
- α = .05
- α = .01
- α = .001
- In this example, assume α = 0.05

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

4. Calculating Test Statistics

Sample mean

4. Calculating Test Statistics

Standard deviation of observations

4. Calculating Test Statistics

Calculated t

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

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.

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

6. Compare the Computed Test Statistic Against a Tabled Value

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

If |tc| > |tα| Reject H0

If p value < α Reject H0

Pvalue is one minus

probability of observing

the t-value calculated

from our sample

Example of Decision Rules

- In terms of t score:

|tc = 2.449| > |tα= 2.262| Reject H0

- In terms of p-value:

If p value = .037 < α = .05 Reject H0

Constructing a Confidence Interval for µ

Standard deviation of sample means

Sample mean

Critical t value

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

Distribution of Mean of Samples

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

will include m

Sample Report of One Sample t-test in Literature

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

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 SPSSSPSS Output: One-Sample Statistics

One-Sample Test

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

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