1 / 38

# Testing Statistical Hypothesis The One Sample t-Test - PowerPoint PPT Presentation

Testing Statistical Hypothesis The One Sample t-Test. Heibatollah Baghi, and Mastee Badii. Parametric and Nonparametric Tests. Parametric tests estimate at least one parameter (in t-test it is population mean)

Related searches for Testing Statistical Hypothesis The One Sample t-Test

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Testing Statistical Hypothesis The One Sample t-Test' - leala

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Testing Statistical HypothesisThe One Sample t-Test

Heibatollah Baghi, and

• 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)

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

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

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

• 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

µ=100

σ=16

n = 64

Sample 1

Sample 2

Sample 3

Etc

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.

• 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

• In this example, assume α = 0.05

• 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.

• 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 Value

If |tc| > |tα| Reject H0

### Decision Rule for P-values Value

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 Value

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

### Take Home Lesson Value

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