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Theorem : For a random sample of size n from a population which has the distribution N (μ, ơ), the
sampling distribution of X is N (μ, ơ/ √ n )
Consequently, Z=
X − μ
√ n
~ N (0, 1)
Central Limit Theorem : For large n (n >30), the sampling distribution of X is approximately N (μ, ơ/ √ n )
for any population with finite standard deviation ơ.
Desired properties of a point estimation : Unbiasedness Small standard error Interval estimators are often in the form: Point estimator +or – margin of error Margin of error : The quantity that is added or subtracted from the point estimator to determine the interval estimator = (table value) X (standard error) Two measures of quality:
o Confidence level (Confidence coefficient): 1 – α ,– the probability that the interval
estimator will yield an interval that includes the parameter Common choices are .90, .95,. o Length of the interval (or the margin of error) Desired property: o Confidence level large and interval length small Setting 1 : Normal population, ơ is known, random sample
¿)^ Z = X^ − μ 0 /( σ^ / √^ n )
Common choices for the confidence level:
1 – α α / 2 zα / 2
NEVER use the word Probability when interpreting an interval estimate, instead use confident
Properties of a 100(1- α )% confidence interval estimator for μ , ¿ )
- As the confidence level increases, the length of the interval increases
o As 1- α increases, α decreases, zα / 2 increases, margin of error increases, and length of
interval increases
- As sample size increases, the length of the interval decreases o As n increases, margin of error decreases, and length of interval increases
- As the population standard deviation increases, the length of the interval increases o As ơ increases, margin of error increases, and length of the interval increases How to determine sample size: First, specify the desired confidence level and margin of error (or interval length ) Solve for n using the margin of error formula
E = z^ α
2
√ n^
= n = ( z^ α
2
X ơ / E ¿
2 o Always round up o If ơ is unknown, use s (sample standard deviation) from a small pilot study Setting 2: Any population, ơ known, random sample, large n
¿ ) Z = X^ − μ^ /(
√ n^ )
Setting 3: Any population, ơ unknown, random sample, large n
¿ ) Z = X^ − μ^ /(
s
√ n )
*** Knowing info on population trumps any other factor
T.S : Z = X^ − μ^ /(
√ n )
- Rejection Region : The numerical values of the test statistic for which the decision is to reject
H 0 ; it contains those values of the test statistic which are most supportive of Ha ; its size is
determined by specifying α
R.R: z > zα =
- Sample value of the test statistic – computed value of the test statistic using the sample data collected
- Decision and conclusion: Conclusion is decision rewritten using the words of the problem p-value : The probability, computed under the condition that the null hypothesis is true, of the test
statistic being at least as extreme (more supportive of Ha ) as the value of the test statistic that was
actually observed OR
The smallest significance level ( α ¿for which the decision will be reject H 0
- Let the sample value of the test statistic be the critical value – the cutoff between the rejection region and the acceptance region Making decisions based on p-value:
If p-value < α , reject H 0
If p-value > α fail to reject H 0
***p-value measures the strength of the evidence supporting Ha , the smaller the stronger
Setting 4: Normal population, ơ unknown, random sample, small n
¿ ) T^ = X^ − μ^ /(
s
√ n^ )
Student’s t distribution:
t α ,df – Value of t distribution with degrees of freedom (df) such that the area to the right is α
o Df= n- Properties:
o Bell shaped o Symmetric about its mean 0 o Standard deviation is greater than 1. o There is a family of student’s t distributions: each member of the family is denoted by an index number called degrees of freedom o As the degrees of freedom increase, the student’s t distribution approaches the standard normal distribution
Setting 5: Paired data, normal population, random sample, small n
¿ ) T^ = D − μ /(
s d
√ n )
