10-2 Estimating a Population Mean (σ Unknown). When we substitute the standard error of xbar for its standard deviation, the distribution of the resulting statistic, t , is not Normal. We call it the t distribution. The t distributions.
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Estimating a Population Mean
We call it the t distribution.The t distributions
We specify a t distribution by giving its degrees of freedom, which is equal to n-1
We will write the t distribution with k degrees of freedom as t(k) for short.
We also will refer to the standard Normal distribution as the z-distribution.
Compare the shape, center, and spread of the t-distribution with the z-distribution.
As the degrees of freedom k increase, the t(k) density curve approaches the N(0,1) curve ever more closely. As the sample size increases, s estimates σ ever more closely.
If you have a TI-84+, you can use invT((1+C)/2, df) to find t*.
Subjects are matched in pairs and each treatment is given to one subject in each pair.Recall matched pairs design...
each subject receives both treatments in some randomized order
When you have two sets of data, ask yourself if there is something that links the values in pairs and, therefore, prevents them from being independent. If so, a one-sample procedure is optimal.
Inference procedures for two samples assume that the samples are selected independently of each other. This assumption does not hold when the same subjects are measured twice.
Too many numbers what do I do?
Okay so it’s the difference in the means for the entire population.
Random Assignment of treatments to subjects in an experiment lets us investigate whether there is evidence of a treatment effect (cause and effect). That is it lets us compare results of different treatments.Randomization
No, since the outlier suggests that the population may not be Normal.One sample t interval for mu
ALWAYS make a plot to check for skewness and outliers BEFORE using t procedures for small samples.Robust Procedures
For most purposes, you can safely use the one-sample t procedures when unless an outlier or some strong skewness is present.Robust Procedures
Hint: This is a population not a sample. Do we want to estimate a parameter when we have a census?Can we use t?
Hint: n =70 and the distribution is what shape?Can we use t?
Hint: n is unknown.Can we use t?