Medical Statistics Medical Statistics Tao Yuchun Tao Yuchun 5

Statistical inference Statistical inference 1. Estimation of population parameter

Sampling error and standard error of mean Sampling Study Sampling error Sampling error: Sample → sample mean (different from population mean) Different samples → Different sample means (different from each other )

(1) Sampling error is related to the variation of the population variation of the population No variation, no error, also no Sampling error! Example Example: The sample means of systolic blood pressure. For adult population (age 25~90) -- vary substantially For young population (age 18~25) -- not vary too much No variation, no statistics, too !

(2) Sampling error is also related to sample size sample size If sample size = population size there is no sampling error! If sample size = 1 Sampling error ≡ variation of population!

How changes for sample mean? See simulative experiment below Sampling from N(4.6602, ) by computer (100 times)

8 Sampling from a skew distribution by computer (a) n=5 (b) n=10 (c) n=20 (d) n=30 (e)

The distribution of sample mean If the variable ~ a normal distribution sample means ~ a normal distribution If the distribution of variable ~ skew, For small sample distribution of sample mean – skew For large sample sample mean close to a normal distribution --Came from Central Limit Theorem

standard error Standard deviation of the population: Standard deviation of the sample mean or Standard error of sample mean or Standard error: In any case: Standard error of sample mean = or

For application S is estimation of σ, is estimation of.

Student’s t distribution The t distribution was discovered William S. Gosset by William S. Gosset in “Student” is his pen name  For a normal distribution William S. Gosset If Z follows a standard normal distribution ---N(0,1).

When σ is unknown, t follows a t distribution. t curve

The Property of t DistributionThe Property of t Distribution I. centrosymmetric Center is 0. II. ν — shape parameter degree of freedom also called degree of freedom, ν = n-1. determine shape of a t curve. different ν, different t curve. When ν is increasing, t curve is close to standard normal curve; when ν →∞, t curve became standard normal curve. See this animation In statistics, the number of degrees of freedomstatistics is the number of values in the final calculation of a statistic that are free to vary. --Came from Wikipedia

The different t curves ν= ∞ (standard normal curve) ν= 4 ν= 1 f(t)f(t)

III. The area under the t curve The Table for t distribution. t value denotes, α is probability, ν is degree of freedom, ν = n-1. The area under the t curve means: One side ： P(t≤-t α,ν )=α or P(t≥t α,ν )=α Two sides ： P(t≤-t α,ν )+P(t≥t α,ν )=α See next figure

ν The meanings of the area under the t curve for two sides

Confidence Interval of Population Mean Statistical inference Estimation parameter Hypothesis testing point estimation interval estimation  Point estimation of population mean -- sample mean  Interval estimation of population mean -- (1-α) confidence interval  Confidence level: 1-α, such as 95% or 99%.

From P(t≤-t α,ν )+P(t≥t α,ν )=α We can get (1- α) confidence intervalIt is the formula of (1- α) confidence interval of population mean of population mean for two sides.

(1- α) confidence interval of population mean95% CI99% CI mean can abbreviate to 95% CI or 99% CI. Whenever we get a mean and standard deviation from a sample, put them into then The two extreme values are called confidence limits.

Example Systolic blood pressures of 20 healthy males were measured. What is 95% confidence interval of the population mean? came from the Table of t distribution

% CI: What does “confidence interval” mean? (1-α) CI Not include μ

C  You should know  You should know: Once you got a 95% confidence interval of the certain population mean, the μ for this population may be in it, also may not be in it, but the probability probability being in it is 95% ! (Guilin Pagodas

In figure, the red curve is standard normal curve ， the blue curve is t curve ， df is ν (degree of freedom).