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Hypothesis Testing of Single Mean and Single Proportion: Assumptions

Module by: Dr. Barbara Illowsky, Susan Dean

When you perform a hypothesis test of a single population mean μμ using a Student-t distribution (often called a t-test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is larger than 30, a t-test will work even if the population is not approximately normally distributed).

When you perform a hypothesis test of a single population mean μμ using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed or your sample size is larger than 30 or both. You know the value of the population standard deviation.

When you perform a hypothesis test of a single population proportion pp, you take a simple random sample from the population. You must meet the conditions for a binomial distribution which are there are a certain number nn of independent trials, each trial has the same probability of a success pp, and the outcomes of any trial are success or failure. The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities npnp and nqnq must both be greater than five (np>5np>5 and nq>5nq>5). Then the binomial distribution of sample (estimated) proportion can be approximated by the normal distribution with μ=npμ=np and σ=npqσ=npq. Remember that q=1-pq=1-p.

Glossary

Binomial Distribution:
A discrete random variable (RV) which arises from the Bernoulli trials with the next additional requirements. There are fixed number, n, of independent trials. “Independent” means that the result to any trial (for example, trial 1) in no way affects the answer to all the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV XX size 12{X} {} is defined as the number of success in n trials. The notation is: XX~ B ( n , p )B(n,p); the domain is the mean is μ=np μ np , and the variance is σ 2 = df σ 2 =df. The probability to have exactly xx successes in nn trials is P ( X = x ) = n x p x q n − x P(X=x)= n x p x q n − x .
Hypothesis Testing:
Based on sample evidence procedure to determine whether the hypothesis stated is a reasonable statement and cannot be rejected, or is unreasonable and should be rejected.
Normal Distribution:
A continuous random variable (RV) with pdf=1σ2πe−(x−μ)2/2σ2pdf=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 μμ is the mean of the distribution and σσ is its standard deviation. Notation: XX ~ N μ σ 2 N μ σ 2 . If μ=0μ=0 and σ=1σ=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 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 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.

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