INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 6: Introduction to Inference

(Part 1)

Dr. Nahid Sultana

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Chapter 6 Introduction to Inference

6.1 Estimating with Confidence

6.2 Tests of Significance

6.3 Use and Abuse of Tests

6.4 Power and Inference as a Decision

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6.1 Estimating with Confidence

Inference

Statistical Confidence

Confidence Intervals

Confidence Interval for a Population Mean

Choosing the Sample Size

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Overview of Inference

Methods for drawing conclusions about a population from sample

data are called statistical inference

Methods: Confidence Intervals - for estimating a value of a population parameter Tests of significance – which assess the evidence for a claim about a population

Both are based on sampling distribution

Both use probabilities based on what happen if we used the inference procedure many times.

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How Statistical Inference Works

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Statistical Estimation

Estimating µ with confidence.

Problem: population with unknown mean, µ

Solution: Estimate µ with x

But does not exactly equal to µ x

How accurately does estimate µ? x

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Since the sample mean is 240.79, we could guess that µ is “somewhere” around 240.79. How close to 240.79 is µ likely to be?

To answer this question, we must ask:

?population the from 16 size ofSRSs many took weif vary mean sample the How would x

Statistical Estimation

.16 size of SRS afor 79240mean sample and ;20 :ondistributi population theSuppose

n.x)N(µ, σ

===

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Statistical Estimation (Cont…)

10. and 10-between be willx all of 95% rule, 99.7–95–68 Using

+◊

µµ

.x of points 10 within is that saying as same theis

of 10 within lies xsay that To µ

µ◊

250.79. 10 x and 230.79 10-xbetween lies mean unknown that theconfident 95% are esay that w We240.79. x Here

=+==

µ

5). ,N( :x ofon Distributi µ

. of 10 within is 240.79x that confidence 95% are weHere

µ=◊

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Confidence Interval

estimate ± margin of error

The sampling distribution of tells us how close to µ the sample mean is likely to be. All confidence intervals we construct will have the form:

x x

The estimate ( in this case) is our guess for the value of the unknown parameter. The margin of error (10 here) 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.

We can choose the confidence level C, but 95% is the standard for most situations. Occasionally, 90% or 99% is used.

We write a 95% confidence level by C = 0.95.

The interval of numbers between the values ± 10 is called a 95% confidence interval for μ.

) 10. x and 10-xbetween lies mean that confident (95% +µ

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Confidence Level The sample mean will vary from sample to sample, but when we use the method estimate ± margin of error to get an interval based on each sample, C% of these intervals capture the unknown population mean µ.

The 95% confidence intervals from 25 SRSs

In a very large number of samples, 95% of the confidence intervals would contain μ.

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Confidence Interval for a Population Mean We will now construct a level C confidence interval for the mean μ of a population when the data are an SRS of size n. The construction is based on the sampling distribution of the sample mean . x This sampling distribution is exactly when the population distribution is N(µ,σ). By the central limit theorem, this sampling distribution is appt. for large samples whenever the population mean and s.d. are μ and σ.

)σN(µ, n/

)σN(µ, n/

Normal curve has probability C between the point z∗ s.d. below the mean and the point z∗ s.d. above the mean.

Normal distribution has probability about 0.95 within ±2 s.d. of its mean.

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Confidence Interval for a Population Mean (Cont…)

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Values of z∗ for many choices of C shown at the bottom of Table D:

Choose an SRS of size n from a population having unknown mean µ and

known standard deviation σ. A level C confidence interval for µ is:

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

nzx σ*±

nzm σ*=

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Confidence Interval for a Population Mean (Cont…)

)59.250,99.230(8.979.24016

2096.179.240*

=±=

⋅±=⋅±n

zx σ

79240mean Sample.16 size of SRS

20 :ondistributi Population

.x n

);N(µ, σ

==

=

Calculate a 95% confidence interval for µ.

nzx σ*±

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Confidence Interval for a Population Mean (Cont…)

Margin of error for the 95% CI for μ: 19803.1981200

3500)960.1(* ≈===n

zm σ

95% CI for μ: )3371 ,2975(1983173 =±=± mx

Example:. Let’s assume that the sample mean of the credit card debt is $3173 and the standard deviation is $3500. But suppose that the sample size is only 300. Compute a 95% confidence interval for µ.

Margin of error for the 95% CI for μ: 396300

3500)960.1(* ===n

zm σ

95% CI for μ: )3569 ,2777(3963173 =±=± mx

Example: A random pool of 1200 loan applicants, attending universities, had their credit card data pulled for analysis. The sample of applicants carried an average credit card balance of $3173. The s.d. for the population of credit card debts is $3500.

Compute a 95% confidence interval for the true mean credit card balance among all undergraduate loan applicants.

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The Margin of Error

How sample size affects the confidence interval. Sample size, n=1200; Margin of error, m= 198 Sample size, n=300; Margin of error, m= 396

n=300 is exactly one-fourth of n=1200. Here we double the margin of error when we reduce the sample size to one-fourth of the original value.

A sample size 4 times as large results in a CI that is half as wide.

CI for µ

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How Confidence Intervals Behave

The confidence level C determines the value of z*. The margin of error also depends on z*.

m = z *σ n

C

z*−z*

m m

The user chooses C, and the margin of error follows from this choice. We would like high confidence and a small margin of error.

To reduce the margin of error: Use a lower level of confidence (smaller C, i.e. smaller z*). Increase the sample size (larger n). Reduce σ.

High confidence says that our method almost always gives correct answers. A small margin of error says that we have pinned down the parameter quite precisely

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How Confidence Intervals Behave Example: Let’s assume that the sample mean of the credit card debt is $3173 and the standard deviation is $3500. Suppose that the sample size is only 1200. Compute a 95% confidence interval for µ.

Margin of error for the 95% CI for μ: 1981200

3500)960.1(* ===n

zm σ

95% CI for μ: )3371,2975(1983173 =±=± mxExample: Compute a 99% confidence interval for µ. Margin of error for the 99% CI for μ: 260

12003500)576.2(* ===

nzm σ

99% CI for μ: )3433,2913(2603173 =±=± mx

The larger the value of C, the wider the interval.

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Impact of sample size

The spread in the sampling distribution of the mean is a function of the number of individuals per sample. The larger the sample size, the smaller the s.d. (spread) of the

sample mean distribution. The spread decreases at a rate equal to √n.

Sample size n Sta

ndar

d de

viat

ion

σ ⁄ √

n

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To obtain a desired margin of error m, plug in the value of σ and the value of z* for your desired confidence level, and solve for the sample size n.

2* *

=⇔=

mzn

nzm σσ

*n

zm σ=

Example: Suppose that we are planning a credit card use survey as before. If we want the margin of error to be $150 with 95% confidence, what sample size n do we need? For 95% confidence, z* = 1.960. Suppose σ = $3500.

209254.2091150

3500*96.1 * 22

≈=

=

=

mzn σ

Would we need a much larger sample size to obtain a margin of error of $100?

Choosing the Sample Size