• a sample from population with parameters μ, σ²

• sample mean – a random variable

• distribution of the sample mean – the probability distribution of all possible values of the random variable

• Variance of the sample mean → σ2/n

• standard deviation of the sample mean → σ/√n

• the sample mean is a point estimate of the population mean μ

Point estimate of the variance and the s. deviation

• s² is considered as a point estimate of the variance

• s is considered as a point estimate of the standard deviation

• The population parameter is estimated with an interval that includes the population parameter with given probability.

• Student’s t Distribution - The t is a family of distributions

The t value depends on degrees of freedom (d.f.)

• Number of observations that are free to vary after sample mean has been calculated

d.f. = n - 1

4 Prezent

• Hypothesis testing is a procedure, based on sample evidence and probability, used to test statement regarding a characteristics of one or more populations

• Generalization of results based on sample upon population

• Statistical hypothesis is a statement regarding

  • a parameter (of one or more populations)

  • or a probability distribution

The null hypothesis

• Is a statement to be tested

• It is always a statement of no difference, always contains a statement of equality

The alternative hypothesis

• an opposite statement to the null hypothesis

• H₁: μ₁≠μ₂

• H₁: μ₁>μ₂

• H₁: μ₁<μ₂

Type I error –rejecting H₀ when H₀ is true

probability of Type I error - the level of significance α

Type II error – not rejecting H₀ when H₁ is true

probability of Type II error → β

1-β…test power

Testing methods:

Parametric: require knowledge or estimation of population parameters (average, variance…)

- presumption: normally distributed population

Nonparametric: do not requiere knowledge, are based on ordering the values

Categories of statistical tests

• 1. One sample tests

• 2. Two sample tests

  • independent observations

  • dependent observations

• 3. Comparison more than 2 samples

  • independent observations

  • dependent observations

• One sample tests

• One sample test for the mean

  • known population variance

  • unknown population variance

• σ² is known

• H₀: μ=μ₀

• σ² is unknown

• H₀: μ=μ₀

• One sample test for proportion

• What is null hypothesis for one sample t-test?

• What is significance level?

• When we use t-test to test mean?

• What is null hypothesis for one sample test for proportion?

• What is the decision rule of the test using p-value?

Two sample tests

• Two sample test for variance

• Two sample test for mean

  • independent samples

  • dependend samples

• Two sample test for proportion

Two sample test for variance (F-test)

Scheme of two-sample test for the mean – independent samples

1Step

F – test

(testing σ²₁=σ²₂)

• If H₀ is not rejected

• (σ²₁=σ²₂)

• 2. Step two-sample t-test

or

• If H₀ is rejected

• (σ²₁>σ²₂)

• 2. Step Welch test

dependent samples – individuals selected for one sample are used to determine the individuals to be in the second sample

• example:

  • left and right hand of a person

  • comparing two methods

  • before-after

• What we test using F-test?

• What is significance level?

• When we use paired t-test instead of two sample t-test?

• What is difference between dependent and independent samples?

• When we use Welch test?

• What is null hypothesis for two sample test for proportion?

Sense of ANOVA