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Hypothesis Testing of Single Mean and Single Proportion: Distribution Needed for Hypothesis Testing

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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Earlier in the course, we discussed sampling distributions. Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a student-t distribution. (Remember, use a student-t distribution when the population standard deviation is unknown and the population from which the sample is taken is normal.) In this chapter we perform tests of a population proportion using a normal distribution (usually nn is large or the sample size is large).

If you are testing a single population mean, the distribution for the test is for averages:

X¯ X ~ N ( μ X , σ X n ) N( μ X , σ X n ) or t df t df

The population parameter is μμ. The estimated value (point estimate) for μμ is x¯x, the sample mean.

If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

P 'P' ~ N ( p , p q n ) N(p, p q n )

The population parameter is pp. The estimated value (point estimate) for pp is p'p'. p'= xnp'=xn where xx is the number of successes and nn is the sample size.


Normal Distribution:
A continuous random variable (RV) with pdf f(x)=1σe(xμ)2/2f(x)=1σe(xμ)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 μμ is the mean of the distribution and σσ is its standard deviation. Notation: XX ~ N μ σ N μ σ . If μ=0μ=0 and σ=1σ=1, the RV is called the standard normal distribution.
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 σσfor population standard deviation.
Student-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:
  • It is continuous and assumes any real values.
  • The pdf is symmetrical about its mean of zero. However, the graph is more spread out and flatter at the apex than the normal distribution.
  • It approaches the standard normal distribution as n gets larger.
  • 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.

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