Chapter7-MoreTesting - CHAPTER 7 MORE CONFIDENCE INTERVALS...

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CHAPTER 7 – MORE CONFIDENCE INTERVALS AND HYPOTHESIS TESTS STAT 200
The Big Picture of Chapter 7 2 Up to now we’ve seen the Z-test and the confidence intervals for means. Both methods required three conditions: Independent observations Normally distributed data (Or large sample size) Standard deviation is known In this chapter we look to overcome the third assumption and to generalize to two sample situations.
One Sample Procedures PART 1 3
t-Confidence Interval 4 In situations where we seek to estimate the mean , we almost never know the standard deviation . In order to overcome this issue in confidence intervals, we may be tempted to simply replace by S – the sample standard deviation used to estimate . This is a correct step, but it is not the only change that we must bring to the interval.
What changes when σ is unknown? Notice that we are now dividing by an estimated standard deviation. This isn’t a Z-score! As a result the distribution of the middle term is not Normal 5
Not so Normal Anymore 6 Since we use S , a random variable in the denominator instead of a fixed value, we add uncertainty to the middle term. This translates into higher variance which is accounted for by the new distribution we use, called the t-distribution. So where does the distribution come into play in all this? It’s from the distribution that we get the confidence coefficients.
t - Distribution If the population is normally distributed and σ is unknown, the appropriate interval is based on the t - distribution The t -Distribution are a family of similar distributions defined by a single parameter called its degrees of freedom (d.f.) . They are symmetric, bell-shaped curves. Basically they are like the standard Normal but with a larger spread (centered at zero with thicker tails ). The larger the degrees of freedom, the closer to the standard Normal it gets and hence it gets less dispersion. 7
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T-Table (Table A.4) 9
Calcium Using specimens obtained from 10 individuals, determinations of percent calcium content of sound teeth gave the following results: x 1 +x 2 +…x 10 : 356.58 s=0.7137 Find 95 and 99 percent CIs for the mean percentage of calcium in the population. 10
Validity of CI’s If the population distribution is Normal and σ is known: the CI based on the normal distribu7on is exact σ is unknown: the CI based on the t‐distribu7on is exact. If the popula7on distribu7on is not Normal, but n is sufficiently large, the CI based on the corresponding t‐distribu7on is approximately correct (due to a theorem similar to the CLT). 11
continued In practice, if n is large, the CI based on the normal distribution with s substituted for σ also is approximately correct since the t-distribution with large df is almost identical to the standard Normal distribution.

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