Hypothesis Testing of Single Mean and Single Proportion: Distribution Needed for Hypothesis Testing 1.4 2008/06/06 17:21:50 GMT-5 2008/10/27 16:34:49.899 GMT-5 Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Connexions cnx@cnx.org elementary statistics 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.) Perform tests of a population proportion using a normal distribution (usually n is large).If you are testing a single population mean, the distribution for the test is for averages: X ~ N ( μ X , σ X n ) or t df (See Chapters 7 and 8)The population parameter is μ. The estimated value (point estimate) for μ is x, the sample mean.If you are testing a single population proportion, the distribution for the test is for proportions or percentages: P ' ~ N ( p , p ⋅ q n ) (See Chapter 8)The population parameter is p. The estimated value (point estimate) for p is p'. p'= xn where x is the number of successes and n is the sample size. Normal Distribution A continuous random variable (RV) with pdf=1σ2πe−(x−μ)2/2σ2 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 its standard deviation. Notation: X ~ N μ σ 2 . If μ=0 and σ=1, the RV is called standard normal distribution, or z-score. Standard Deviation A number that is equal to the square root of the variance and measures how far data values are from their mean. Notations: s for sample standard deviation and σfor 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 a continuous and assumes any real values. The pdf is symmetrical about its mean of zero. However, it 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 family is completely defined by the number of degrees of freedom which is one less than the number of data.