Chapter 6

Statistical inference draws conclusions about a population or process based on sample data

Two types of statistical Inferences:

-Confidence interval

-Tests of significance

This chapter will only consider scenarios of inference about the mean with the standard deviation

given.

In this setting we can ask questions like:

-What is the average loan debt among undergraduate borrowers?

-What is the average mile per gallon (mpg) for a hybrid car?

- Is the average cholesterol level of undergraduate women at your university below the

national average?

Overview of inference

Purpose of statistical inference is to draw conclusion from data.

Probability allows us to take chance variation into account

Ex 6.2 Effectiveness of a new drug

20 were given the drug 60% had improvement

20 were given the placebo 40% had improvement

Due to probability calculations a difference this large or larger between the results in the two

group would occur one time in five simply because of chance variation.

Confidence interval – used for estimating the value of a population parameter

Tests of significance – assess the evidence for a claim

Both types of inferences are based on sampling distributions of statistics

Probability models are most secure, and inference is most reliable when the data are produced by

a properly randomized design.

Unrealistic assumption: is when the standard deviation ( ) is given.

6.1 Estimating with Confidence

Sample mean (x-bar) can be used for mean (µ) of population since it is unbiased

But each sample mean (x-bar) can differ from sample to sample

An estimate without an indication of its variability is of little value.

Statistical confidence

n = 500

= 100

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x = 100/ square root 500 = 4.5

Consider:

-The 68-95-99.7 rule says that the probability is about 0.95 that x-bar will be within 0

points (2 of the population mean score µ.

-To say that x-bar lies within 9 points of µ is the same as saying that µis within 9 points of

x-bar

-So 95% of all samples will capture the true µ in the interval from x-bar – 9 to x-bar + 9

The language of statistical inference uses this fact about what would happen in the long run to

express our confidence in the results of any one sample.

X-bar = 461

95% confidence lies between: 452 and 470

X-bar – 9 = 461 – 9 = 452 and X-bar + 9 = 461 + 9 = 470

2 possibilities of our SRS:

1. The interval between 452 and 470 contains the true µ.

2. The interval between 452 and 470 does not contain the true µ.

95% confidence means that the method used gives correct result 95% of the time.

Confidence Intervals

The interval of # between the values X-bar+9 is called the 95% confidence interval for µ.

X-bar+9 = Estimate + margin of error

Margin of error reflects how accurate we believe our guess is based on the variability of the

estimate, and how confident we are that the procedure will catch the true population mean µ.

2 important things about confident intervals

1. It is an interval of the form (a,b), where a and b are numbers computed from the data.

2. It has a property called a confidence level that gives the probability of producing an

interval that contains the unknown parameter.

Occasionally, 90% and 99% is used, but the 95% rule is mostly used.

C stands for confidence level

A confidence of 95% C= 0.95

Confidence Interval

A level C confidence interval for a parameter is an interval calculated from a sample data be a

method that has a probability C of producing an interval containing the true value of the

parameter.

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Ex 6.4 80% confidence intervals

Confidence level tells us what percent will capture µ in the long run.

As number of samples increases, the percent of captures gets closer to the confidence level, 80%.

Confidence interval for a population mean

Sampling distribution of the sample mean x-bar is exactly N(µ, /square root n)

Any normal distribution has probability about 0.95 within + 2 x-bar

C = confidence interval

Z* = # + the µ value

Normal distribution has probability c within + z

Ex: C = 90%, z* = 1.645

C = 95%, z* = 1.96 most important entries

C = 99%, z* = 2.576

Z* values are found at the bottom of table D

For the probability C that X-bar lies between

µ - z* ( /square root n) and µ + z* ( /square root n)

same as saying that the unknown population mean µ lies between

x-bar – z*( /square root n) and x-bar + z*( /square root n)

In other words there is a probability C that the interval x-bar = z*/square root n , contains µ.

The estimate of the unknown µ is x-bar, and the margin of error is z*/square root n

Confidence interval for a population mean

Choose an SRS of a size of population having unknown mean µ and known standard deviation .

The margin of error for a level C confidence interval for µ is

m = z* ( /square root n)

Here z* is the value on the standard Normal curve with area C between the critical points –z*

and z*. The level C confidence Interval for µ is

x-bar + m

This interval is exact when the population distribution is Normal and is approximately correct

when n is large in other cases.

Ex 6.4 Average debt of undergraduate borrowers

Given

n = 1280

x-bar = $18900, = $49000

C = 0.95 use table d to find z*

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