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H o and H a are contradictory.

If H o has: equal (=) greater than or equal to in problem statement still translates to equal to ( = ) less than or equal to in problem statement still translates to equal to ( = )
then H a has: not equal (≠) or greater than ( > ) or less than ( < ) less than ( < ) greater than ( > )

If α p-value, then do not reject H o .

If α > p-value, then reject H o .

α is preconceived. Its value is set before the hypothesis test starts.The p-value is calculated from the data.

α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true.

β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false.

If there is no given preconceived α , then use α = 0.05 .

    Types of hypothesis tests

  • Single population mean, known population variance (or standard deviation): Normal test .
  • Single population mean, unknown population variance (or standard deviation): Student's-t test .
  • Single population proportion: Normal test .

Glossary

Binomial distribution

A discrete random variable (RV) which arises from Bernoulli trials. There are a fixed number, n , of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X size 12{X} {} is defined as the number of successes in n trials. The notation is: X ~ B ( n , p ) . The mean is μ np and the standard deviation is σ = npq . The probability of exactly x successes in n trials is P ( X = x ) = n x p x q n x .

Central limit theorem

Given a random variable (RV) with known mean μ and known standard deviation σ. We are sampling with size n and we are interested in two new RVs - the sample mean, x ¯ size 12{ {overline {x}} } {} , and the sample sum, ΣX . If the size n of the sample is sufficiently large, then X ~ N ( μ X , σ X n ) and ΣX ~ N(nμ, n σ). If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σ n is called the standard error of the mean.

Confidence interval (ci)

An interval estimate for an unknown population parameter. This depends on:
(1). The desired confidence level.
(2). Information that is known about the distribution (for example, known standard deviation).
(3). The sample and its size.

Hypothesis

A statement about the value of a population parameter. In case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation H o ) and the contradictory statement is called the alternate hypothesis (notation H a ).

Hypothesis testing

Based on sample evidence, a procedure to determine whether the hypothesis stated is a reasonable statement and cannot be rejected, or is unreasonable and should be rejected.

Level of significance test

Probability of a Type I error (reject the null hypothesis when it is true). Notation: α. In hypothesis testing, the Level of Significance is called the preconceived α or the preset α.

Normal distribution

A continuous random variable (RV) with pdf f(x) = 1 σ e ( x μ ) 2 / 2 size 12{ ital "pdf"= { {1} over {σ sqrt {2π} } } e rSup { size 8{ - \( x - μ \) rSup { size 6{2} } /2σ rSup { size 6{2} } } } } {} , where μ is the mean of the distribution and σ is the standard deviation. Notation: X ~ N μ σ . If μ = 0 and σ = 1 , the RV is called the standard normal distribution .

P-value

The probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.

Standard deviation

A number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation and σ for population standard deviation.

Student's-t distribution

Investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:
(1). It is continuous and assumes any real values.
(2). The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
(3). It approaches the standard normal distribution as n gets larger.
(4). There is a "family" of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data.

Type 1 error

The decision is to reject the Null hypothesis when, in fact, the Null hypothesis is true.

Type 2 error

The decision is to not reject the Null hypothesis when, in fact, the Null hypothesis is false.

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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