parametric statistical inference

PARAMETRIC STATISTICAL INFERENCE

INFERENCE:• Methodologies that allow us to draw

conclusions about population parameters from sample statistics

TYPES OF INFERENCE: 1. Estimation2. Hypothesis testing

• Methods based on statistical relationships between samples and populations

• POINT ESTIMATION: estimation of parameter from a sample statistic

– For the mean, standard deviation, etc..

• INTERVAL ESTIMATION: using a sample to identify an interval within which the population parameter is thought to lie, with a certain probability

ESTIMATION OF POPULATION MEAN

• Sample mean value is only an estimate of the parameter mean value

– Parameter value is not known

• Due to sampling variability, no two samples will produce exactly the same outcome, or sample mean

Can we estimate how this sample mean

value would vary if you take many large samples from the same population?

Remember: sample mean values from large samples

have a normal distribution the mean of the sampling distribution is

the same as the unknown parameter • standard deviation of for a SRS of size n

is ? x

PARAMETRIC STATISTICAL INFERENCE: ESTIMATION

• Example: A random sample of 350 male college students were asked for the number of units they were taking. The mean was 12.3 units, with a standard deviation of 2.50 units.

• What can we say about the mean

number of units of all student males at the university? How will the estimate value of the parameter vary from one sample to another with a certain confidence, like 95%?

Assume that = ?. s = ?


Statistical confidence Remember: The 68-95-99.7 rule In 95% of all samples, the mean score of

x will lie within 2 standard deviations of the population mean score .

Since s = 2.50, we can say that In 95% of samples, will lie within 5.0

points of the observed sample mean In 95% of all samples,

• Thus, the parameter will lie between 7.3 and 17.3, in 95% of samples

0.50.5 xx


Rephrasing: 1. We are 95% confident that the interval

7.3-17.3 contains • We have just assigned statistical

confidence to our estimation of the parameter

• We call this estimated interval a CONFIDENCE INTERVAL for the mean value


But, there is still some chance that the true parameter value will not lie in the identified interval

• e.g. The SRS chosen was one of few samples for which is not within 5.0 points of true mean. 5% of samples will give these incorrect results

x


CONFIDENCE INTERVAL – formal definition

A level C confidence interval for a parameter

is defined as

estimate margin of error and gives the interval that will capture the

true parameter value in repeated samples with a certain probability

Confidence intervals usually vary between 90% and 99.9%


Constructing confidence intervals

When we construct a 95% confidence interval, we are looking for two values for which there is a 95% chance that the population mean is between them. So,

P(Low < < High) = 0.95

Thus, 0.95 = P(-1.96 < z < 1.96)

=

=

=

0.95 =

)96.196.1(

n

xP

)96.196.1(n

xn

P

)96.196.1(n

xn

xP

)96.196.1(n

xn

xP


Draw a SRS of size n from a population having unknown mean , and known standard deviation . A level C confidence interval for

This interval is exact when the population distribution is normal and is approximately correct for large n in other cases

Z* = critical value of z – table C

nzx

*

nzx

nzx

**

Figure 6.5 and figure 6.6

C = chosen confidence level – probability that a parameter will lie within a given interval with a desiredconfidence

(1-C)/2 = probability that a parameter will be situatedeither above or below the the lower confidence limit

Confidence intervals and confidence levels of Standardized normal curve N(0,1)

parametric statistical inference

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