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Testing Statistical Hypothesis The One Sample t-Test

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Testing Statistical HypothesisThe One Sample t-Test

Heibatollah Baghi, and

Mastee Badii

- 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

- 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

- 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)

True

False

True

(Accept Ho)

Type II error

Probability = b

False

(Rejects Ho)

Type I error

Probability = a

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

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

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

- 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

- 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.

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

This lecture focuses on

one sample t-test

The One Sample t – Test

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

- 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

- 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

- If sample size is more than 30 use z-test
- If sample size is less than 30 use t-test
- Sample size of 10

- α is a predetermined value
- The convention
- α = .05
- α = .01
- α = .001

- H0 :µ = 6.75
- Ha :µ ≠ 6.75

A two tailed

test because it

can be either larger

or smaller

Sample mean

Deviation

from sample

mean

Squared

deviation

from

sample

mean

Standard deviation of observations

Calculated

t value

Standard deviation

of sample means

Calculated t

- 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

- Suppose you have a sample of three observations:

X

--------

--------

--------

- 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.

- Are these three deviations independent of one another?

- 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

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

Decision Rule for t-Scores

If |tc| > |tα| Reject H0

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

- 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

Standard deviation of sample means

Sample mean

Critical t value

- 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

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

will include m

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

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

SPSS Output: One-Sample Statistics

One-Sample Test

Take Home Lesson

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