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.
H₀the null hypothesis states that the treatment has no effect.

Table of Contents.

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

t-test : ONE Sample

Calculate the test statistic

Calculate the sample mean

t-test : ONE Sample

Calculate the sample standard deviation.

t-test : ONE Sample

Compute the estimated standard error.

Calculate the t statistic for the sample data.

Take a decision regarding H₀

If obtained t_cal statistics fall into the critical region of the t distribution. The decision is to reject H₀
If obtained t_cal 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 H_0. 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

t-test : ONE Sample

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

What is a Hypothesis t-test for ONE sample?

t statistics for unknown Population

Steps of Hypothesis Testing

State the hypothesis

Select an alpha level

Locate the critical region

Calculate the test statistic

Take a decision regarding H0

Assumptions for conducting a t-test

Effect of Sample Variance

Important points: t statistics

t Statistics – Treatment Effect measurement

Evaluating the size of a treatment effect: Cohen d

Percentage of variance

Confidence Intervals for Estimating µ

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Take a decision regarding H₀