Understanding Sampling Distributions & t-Tests: Confidence Intervals & Hypothesis Tests - , Study notes of Business Statistics

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Chapter 11: Confidence Intervals and Hypothesis Tests for Means

 Standard deviation of the sampling distribution is s/√n, measures how much the sample statistic varies from sample to sample. It is smaller than the standard deviation of the population by a factor of √ n. Averages are less variable than individual observations.  The mean of the sampling distribution is an unbiased estimate of the population mean μ. Mean of all



x is pretty much equal to population mean μ.

 Central Limit Theorem: When randomly sampling from any population with mean μ and standard deviation s , when n is large enough, the sampling distribution is approximately normal: ~ N ( μ , s /√ n ). Usually 25, 40 to overcome extreme skewness/outliers. o The mean of a random sample has a sampling distribution whose shape can be approximated by the Normal model; a larger sample will yield a better approximation.  X-bar: sample mean  Standard Error of (X-bar) = SE(X-bar) = s/√n Confidence Intervals  X-bar ± t*n-1 × SE(Y-bar) o same thing as: X-bar +/- ME  For a sample of size n, the sample standard deviation s is

   

( )^2 1 1 x x n s (^) i n – 1 = degrees of freedom (SO much easier to plug into your calculator though... STAT > Edit > Put into L1. Go back into STAT > Calc > 1-Var Stats)  SEM = s /√ n, standard error of the mean T Distributions  One-sample t statistic



t 

x  

s n

 [When s is known, the sampling distribution is N ( m , s /√ n )]  But when s is estimated from the sample standard deviation s , the sampling distribution follows a t distribution t ( μ , s /√ n ) with degrees of freedom n – 1  When n is very large, s is a very good estimate of s , and the corresponding t distributions are very close to the normal distribution.  The t distributions become wider for smaller sample sizes, reflecting the lack of precision in estimating s from s. One-sample T-Confidence Intervals  Margin of error m = t ∗ s^ /√ n^ , leading to xbar +/- the ME One-sample T-Test  Stating the null and alternative hypotheses ( H 0 versus H a)  Deciding on a one-sided or two-sided test  Choosing a significance level a  Calculating t and its degrees of freedom:



t 

x  

s n

 Finding the area under the curve with Table T

 Stating the P-value and interpreting the result The t procedures are robust to small deviations from normality – the results will not be affected too much. Factors that strongly matter:  Random sampling: the sample must be an SRS from the population.  Outliers and Skewedness: they strongly influence the mean and therefore the t procedures. However, their impact diminishes as the sample size gets larger because of the Central Limit Theorem.  Specifically: o When n < 15, the data must be close to normal and without outliers. o When 15 < n < 40, mild skewness is acceptable but not outliers. o When n > 40, the t- statistic will be valid even with strong skewness. Calculating Sample Size m = z ∗ σ

√ n^

⇔ n =

(

z ∗ σ

m )

2 If less than 60, use T