PS296 Chapter Notes - Chapter 12: Variance, Standard Deviation, Sampling Distribution
Testing a Sample Mean When σ is Unknown (The One-Sample t Test)
-we rarely know the value of σ and usually have to estimate it by way of the sample standard
deviation
-when we replace σ with with in the formula z score, the nature of the test changesand we can no
longer declare the answer a z score/evaluate it with reference to z table
-instead we denote the answer as t and evaluate it with respect to tables of t
-we use t to test the hypothesis that a sample came from a population with a specific mean within
the context of not knowing the population standard deviation
The Sampling Distribution of s²
-t test uses s² as an estimate of σ² thus its important we first look at the sampling distriution of s²
-s² is an unbiased estimate of σ², meaning with repeated sampling the average value of s² will equal
σ²
-problem is that the shape of the sampling distribution of s² is quite posivitively skewed esp for
small sample sizes and thus an individual value of s² is more likely to underestimate σ² than to
overestimate it, especially for small samples
-s² is still unbiased bc when it overestimates σ² it does so to such an extent as to balance off the
more numerous but less drastic underestimates
-as a result of skewness the resulting value of t, if we were to just take our sample estimate s² and
substitute for the unknown σ², is that t would be larger than the value of z we would have obtained
had σ² been known
-this is bc any one sample variance s² has a better than 50:50 chance of underestimating σ²
Document Summary
Testing a sample mean when is unknown (the one-sample t test) We rarely know the value of and usually have to estimate it by way of the sample standard deviation. When we replace with with in the formula z score, the nature of the test changesand we can no longer declare the answer a z score/evaluate it with reference to z table. Instead we denote the answer as t and evaluate it with respect to tables of t. We use t to test the hypothesis that a sample came from a population with a specific mean within the context of not knowing the population standard deviation. T test uses s as an estimate of thus its important we first look at the sampling distriution of s . S is an unbiased estimate of , meaning with repeated sampling the average value of s will equal.
