t-test : ONE Sample

What is a Hypothesis t-test for ONE sample?

  • A hypothesis test determines whether the treatment effect is greater than chance, where “chance” is measured by the standard error.
  • H0 the null hypothesis states that the treatment has no effect.

  • The null hypothesis provides a specific value for the unknown population mean.

t statistics for unknown Population

  • t statistics permit hypothesis testing in situations for which population mean is not known to serve as a standard.
  • t-test does not require any prior knowledge about the population mean or the population variance
  • Therefore, a t-test can be used in situations for which the null hypothesis is obtained from a theory, a logical prediction, or thinking.

  • For example, in many studies questions are used to measure perceptions or attitudes. Participants are asked to provide their opinion on questions asked, on a scale from 1 to 10 rating.
        • 1 – Strongly disagree
        • 5 – Neutral
        • 10 – Strongly agree
  • On a scale of 5- The null hypothesis would state that there is no preference, or no strong opinion, in the population, and use a null hypothesis of H0: μ = 5.
  • The data from a sample is then used to evaluate the hypothesis. There is no prior information about the population mean.

Steps of Hypothesis Testing

State the hypothesis

    • H0 : μ = 5 (from the above example)
    • H1 : μ ≠ 5

Select an alpha level

    • Set the level of significance at α = .05 for two tails

Locate the critical region

    • Calculate degrees of freedom df = n – 1
    • For example: sample size =10, df=10-1 = 9
    • The critical region consists of t values greater than +2.262 or less than –2.262 for two tails at α = .05 significance level

Calculate the test statistic

  • Calculate the sample mean

  • Calculate the sample standard deviation.

  • Compute the estimated standard error.

  • Calculate the t statistic for the sample data.

Take a decision regarding H0

  • If obtained tcal statistics fall into the critical region of the t distribution. The decision is to reject H0
  • If obtained tcal statistics fall outside the critical region of the t distribution. Reject H0 & accept H1

Assumptions for conducting a t-test

  • The values in the sample must consist of independent observations.

(Events are independent if the occurrence of the first event has no effect on the probability of the second event.)

  • The population sampled must be normal.
    1. With very small samples, normal population distribution is important.
    2. If in doubt that the population distribution is not normal, use a large sample to be safe.

Effect of Sample Variance

  • Estimated standard error, appears in the denominator of the formula, a larger value for Estimated standard error produces a smaller t value (closer to zero).

Factors that influence standard error⇒

    • Sample standard deviation
      • Larger the standard deviation larger the error.
    • Sample size, n.
      • The larger the sample, the smaller the error
      • large samples tend to produce bigger t statistics and therefore are more likely to produce significant results

Important points: t statistics

  • Large variance is bad for inferential statistics.
  • Large variance is due to Sample data being scattered. It makes it difficult to get patterns or trends in the data.
  • High variance reduces the likelihood of rejecting the null hypothesis H0. And in most cases, the hypothesis result will be derived as no difference/effect after treatment.
  • A large sample size strengthens the likelihood of rejecting the null hypothesis H0. Hypothesis results will be derived as significant differences/effects after treatment.
  • Variance is the behaviour of the population and reflects in samples drawn (variance). We can only control the sample size n.
  • To achieve better Hypothesis analysis results, it is advised to take a larger sample size.

t Statistics – Treatment Effect measurement

  • A hypothesis test simply determines whether the treatment effect is greater than chance, where “chance” is measured by the standard error.
  • A hypothesis test does not evaluate the size of the treatment effect.
  • It is possible for a very small treatment effect to be “statistically significant,” when the sample size is very large.
  • To correct the above two problems, effect size estimation is done with Cohen’s d formula.
  • Cohen defined this measure of effect size in terms of the population mean difference and the “population standard deviation “. It is also called estimated d

Evaluating the size of a treatment effect: Cohen d

Percentage of variance

It is identified as r2.

  • Cohen proposed criteria for evaluating the size of a treatment effect that is measured by r2.
  • Cohen’s standards for interpreting r2 are shown in below Table

Confidence Intervals for Estimating µ

  • A confidence interval is an interval of values centred around a sample statistic.
  • The distribution of t statistics at α = 0.05 for df = 9. The t values pile up around t = 0 and 80% of all the possible values are located between t = –2.262 and t = + 2.262.

where-

    • s represents the sample and
    • p represents the population.
  • The first statement provides the results of the inferential statistical analysis. With degrees of freedom immediately after the symbol t.
  • Effect size is reported by r2
  • After the results of the hypothesis test, the confidence interval is included in the report as a description of the effect size

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