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Testing Statistical Hypothesis The One Sample t-Test






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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)
Testing Statistical Hypothesis The One Sample t-Test

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Slide 1

Testing Statistical HypothesisThe One Sample t-Test

Heibatollah Baghi, and

Mastee Badii

Slide 2

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

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

    • Z-test

    • T-test

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

    • Sample size of 10

Slide 16

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

  • 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

    • 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

    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


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