CHAPTER 17–THINKING ABOUT INFERENCE

17.1 - Conditions for inference about a mean

  • We need a simple random sample
  • The population must be MUCH larger than the sample, say at least 20 times as large
  • Shape of population
  • A normal population is not needed for large samples – ( CLT says that x-bar is approximately normal for large n)
  • For small samples, it is enough that the distribution be symmetric and single peaked unless the sample is very small.
  • Both, mu and sigma are unknown parameters

17.2 - The t-distribution

  • Symmetric about the mean
  • The distribution depends on the degrees of freedom – DF = n - 1
  • Bell shaped, but thicker tails and lower in the middle when you compare it with the z-distribution (SND)
  • As the degrees of freedom increase, the t distribution approaches the standard normal distribution (SND)
  • Mean is zero
  • Standard deviation is more than 1
  • T statistic is
  • Finding t-scores using the t-table
  • For constructing confidence intervals
  • For testing hypothesis

17.3 - The one-sample t-confidence interval

  • The interval is exact when the population is normal and is approximately correct for large n in other cases
  • Construct with the calculator 8:T Interval
  • Construct by hand using the formula:
  • Interpret the results

17.4 - The one-sample t-test

  • Write hypothesis
  • Sketch graph, label and shade
  • Run the test in the calculator 2:TTest
  • Use the results to write the conclusion

By hand:

  • Find the test statistic and use the t-table to find an interval for the p-value
  • Write the conclusion

17.5 Using technology

17.6 – Matched pairs – t-procedures

  • Used to compare responses to the two treatments in a matched pairs design
  • In some cases, each subject receives both treatments in a random order
  • In others, the subjects are matched in pairs as closely as possible, and each subject in a pair receives one of the treatments.
  • Find the difference between the responses within each pair
  • Apply the one-sample t procedures to these differences
  • Read example 17.4 page 455

17.7 – Robustness of the t-procedure

A confidence interval or significance test is called robust if the confidence level or P-value does not change very much when the conditions for use of the procedure are violated.

  • Except in the case of small samples, the condition that the data are an SRS from the population of interest is more important than the condition that the populationdistribution is Normal.
  • Sample size less than 15: Use t procedures if the data appear close to Normal (roughly symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t.
  • Sample size at least 15: The t procedures can be used except in the presence of outliers or strong skewness.
  • Large samples: The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 40.

SUMMARY

  • Tests and confidence intervals for the meanµ of a Normal population are based on the samplemeanof an SRS. Because of the central limit theorem, the resulting procedures are approximately correct for other populationdistributions when the sample is large.
  • The standardized samplemean is the one-sample zstatistic

If we knew σ, we would use the zstatistic and the standard Normal distribution.

  • In practice, we do not know σ. Replace the standard deviationof by the standard errorto get the one-sample tstatistic

The tstatistic has the tdistribution with n − 1 degrees of freedom.

  • There is a tdistribution for every positive degrees of freedom. All are symmetric distributions similar in shape to the standard Normal distribution. The tdistribution approaches the N(0, 1) distribution as the degrees of freedom increase.
  • A level Cconfidence interval for the meanµ of a Normal population is

The critical valuet* is chosen so that the t curve with n – 1 degrees of freedom has area C between –t* and t*.

  • Significance tests for H0: µ = µ0 are based on the tstatistic. Use P-values or fixed significance levels from the t(n − 1) distribution.
  • Use these one-sample procedures to analyze matched pairs data by first taking the difference within each matched pair to produce a single sample.
  • The t procedures are quite robust when the population is non-Normal, especially for larger sample sizes. The t procedures are useful for non-Normal data when n ≥ 15 unless the data show outliers or strong skewness.