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Testing Statistical Hypothesis The One Sample t-Test


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/ratioNonparametric tests do not test hypothesis about specific population parametersDistribution-

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Testing Statistical Hypothesis The One Sample t-Test

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Testing statistical hypothesis the one sample t test l.jpg

Testing Statistical HypothesisThe One Sample t-Test

Heibatollah Baghi, and

Mastee Badii


Parametric and nonparametric tests l.jpg

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 tests3 l.jpg

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


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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)


Types of error in hypothesis testing ho hand washing has no effect on bacteria counts l.jpg

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


Types of error in hypothesis testing ho hand washing has no effect on bacteria counts6 l.jpg

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


Power confidence level l.jpg

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


Level of significance l.jpg

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


Sampling distribution of means l.jpg

Population of IQ scores, 10-year olds

µ=100

σ=16

n = 64

Sample 1

Sample 2

Sample 3

Etc

Sampling Distribution Of Means


Sampling distribution of means10 l.jpg

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.


One sample test l.jpg

One Sample Test

  • 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 l.jpg

The One Sample t – Test

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


An example problem l.jpg

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


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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 l.jpg

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 l.jpg

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 l.jpg

    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 l.jpg

    4. Calculating Test Statistics

    Sample mean


    4 calculating test statistics19 l.jpg

    4. Calculating Test Statistics

    Deviation

    from sample

    mean


    4 calculating test statistics20 l.jpg

    4. Calculating Test Statistics

    Squared

    deviation

    from

    sample

    mean


    4 calculating test statistics21 l.jpg

    4. Calculating Test Statistics

    Standard deviation of observations


    4 calculating test statistics22 l.jpg

    4. Calculating Test Statistics

    Calculated

    t value


    4 calculating test statistics23 l.jpg

    4. Calculating Test Statistics

    Standard deviation

    of sample means


    4 calculating test statistics24 l.jpg

    4. Calculating Test Statistics

    Calculated t


    5 determine degrees of freedom l.jpg

    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 l.jpg

    Degrees of Freedom

    • Suppose you have a sample of three observations:

    X

    --------

    --------

    --------


    Degrees of freedom27 l.jpg

    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 l.jpg

    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 l.jpg

    6. Compare the Computed Test Statistic Against a Tabled Value

    • α = .05

    • Df = n-1 = 9

    • Therefore, reject H0


    Decision rule for t scores l.jpg

    Decision Rule for t-Scores

    If |tc| > |tα| Reject H0


    Decision rule for p values l.jpg

    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


    Example of decision rules l.jpg

    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 l.jpg

    Constructing a Confidence Interval for µ

    Standard deviation of sample means

    Sample mean

    Critical t value


    Constructing a confidence interval for for the example l.jpg

    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 l.jpg

    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 l.jpg

    Sample Report of One Sample t-test in Literature

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


    Testing statistical hypothesis with spss l.jpg

    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


    Take home lesson l.jpg

    Take Home Lesson

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