CH10 Statistical inference -acquire information about populations from samples.
*Point Estimator-A point estimator draws inference about a population by
estimating the value of an unknown parameter using a single value or point.
a) (single value –P(x)=0) virtually certain that the estimate will be wrong.
b) often need to know how close the estimator is to the parameter.
c) (sample n , Not reflect to Parameter value) intuitively reasonable to expect
that a large sample will produce more accurate results, but point estimators don’t Type I Error Reject H0(null hypothesis) when H0is true. P(Type I error)
have the capacity to reflect the effects of larger sample sizes. = α (significance level)- when an innocent person is wrongly convicted
*Interval estimator- An interval estimator draws inferences about a population by Type II Error Do not reject H when H is false. P(Type II error) =
estimating the value of an unknown parameter using an interval.( range of values) β(Greek letter beta)- when a guilty defendant is acquitted
Chapter 11- Hypotheses testing and Inversely related- any attempt to reduce one will increase the other.
If the confidence is 95% then the significance is 5% (alpha). Increasing significance level, |
Four commonly used confidence levels Increasing sample size n, n
The power of a test is defined as 1– P(reject null hypothesis when it’s
Probability of Type II Error:
E.g. = 5%, n = 100, = 10, H : = 200, H : # 200, given = 203
Estimating the Population Mean R.R.: x x 200 x 198.04
X Z Z / 2 1.96
The width of the confidence interval is affected by / 2 / n 10 / 100
• the population standard deviation σ) –SD, confidence interval n
• the confidence level (1-α) –confidence level, confidence interval x x 200 x 201.96
Z /2 1.96
• the sample size (n). –sample size decreases, the confidence interval. / n 10 / 100
Unbiased Estimator-a population parameter is an estimator whose expected P(198 .04 x 201 .96, given 203)
value is equal to that parameter.X)
Consistency- if the difference between the estimator and the parameter grows P(4.96 Z 1.04) .5 .3508 .1492
smaller as the sample size grows larger. ( /n , n, variance) -4.96 -1.04
198 .04 203 201 .96 203
Relative Efficiency-If two unbiased estimators of a parameter, the one whose P ( Z )
variance is smaller is said to be relatively more efficient. (Smaller variance is better)10 / 100 10 / 100
Selecting the Sample size 2 Chapter 12: Inference about a Population when is unknown use t-test
-estimate the mean to within B units X B Z /2
n T distribution (σ is unknown, we use its point estimator s) mound-shaped,
Chapter 11 Hypothesis Test: to determine whether there is enough statistical symmetrical around zero.
The “degrees of freedom” v/ d.f. =n-1, (a function of the samples)ze
evidence in favor of a certain belief about a parameter determine how spread the distribution is (compared to the normal distribution)
Hypothesis -Assume null H 0s true, find evident to infer al1. H T-test only valid when the histogram is NOT extremely nonnormal.
– Null Hypothesis (always “=“)
– Alternative or research Hypothesis (Can be “”, or “=“) Test Statistic for μ When σ Is Unknown
• Test Statistic z x t
/ n s / n
Confidence interval estimating μ when σ is unknown
• Rejection Region -a range of values such that if the test statistic falls into that
range, we decide to reject the null hypothesis in favor of the alternative hypothX t S d.f.=n1
tail RR Z -RR t -RR X -RR / 2 n
2 2 2
> 1(+) right Z> Z t >t ,n 1 X > X ,n 1 Inference about a Population Variance σ
< 1(-) left Z< t < 2 2 The sample variance s is an unbiased, consistent and efficient point estimator
Z t ,n 1 X < X 1 ,n 1 for σ2
= 2( ) both Z> t > 2 2 Test Statistic for σ -Chisquared,
Z / 2 t / 2 ,n 1 X > X / 2 ,n 1 2
Z< Z t < X < X 2 2 (n 1)s d.f.=n1
/ 2 t / 2 ,n 1 1 ( / 2 ),n 1 2
• P-Value (only work in Z distribution)-the probability of observing a test statistic
at least as extreme as the one computed given that the null hypothesis is true. P ( X 21 ( / 2 ) 2 X 2 / 2 1
Eg: The probability of observing a sample mean at least as large as 178 when =170.
(n 1)s 2 2 (n 1)s 2
/ 2 ,n 1 1 / 2 ,n 1
Inference about a Population Proportion (nomin