Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size

1. INTRODUCTION TO
STATISTICS & PROBABILITY
Chapter 6: Introduction to Inference
(Part 1)
Dr. Nahid Sultana
1

2. 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
2

3. 3
6.1 Estimating with Confidence
 Inference
 Statistical Confidence
 Confidence Intervals
 Confidence Interval for a Population Mean
 Choosing the Sample Size

4. 4
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.

5. 5
How Statistical Inference Works

6. 6
Statistical Estimation
Estimating µ with confidence.
Problem: population with unknown mean, µ
Solution: Estimate µ with x
 But does not exactly equal to µx
 How accurately does estimate µ?x

7. 7
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:
?populationthefrom16sizeof
SRSsmanytookweifvarymeansampletheHow would x
Statistical Estimation
.16sizeofSRSafor79240meansampleand
;20:ondistributipopulationtheSuppose
n.x
)N(µ, σ
==
=

8. 8
Statistical Estimation (Cont…)
10.and10-betweenbewillxall
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µ=
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9. 9
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:
xx
 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.xand10-xbetweenliesmeanthatconfident(95% +µ

10. 10
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.

11. 11
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.

12. 12
Confidence Interval for a Population
Mean (Cont…)
12
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
n
zx
σ
*±
n
zm
σ
*=

13. 13
Confidence Interval for a Population
Mean (Cont…)
)59.250,99.230(8.979.240
16
20
96.179.240*
=±=
⋅±=⋅±
n
zx
σ
79240meanSample
.16sizeofSRS
20:ondistributiPopulation
.x
n
);N(µ, σ
=
=
=
Calculate a 95% confidence interval for µ.
n
zx
σ
*±

14. 14
Confidence Interval for a Population Mean (Cont…)
Margin of error for the 95% CI for μ: 19803.198
1200
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: 396
300
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.

15. 15
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

16. 16
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

17. 17
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: 198
1200
3500
)960.1(* ===
n
zm
σ
95% CI for μ: )3371,2975(1983173 =±=± mx
Example: Compute a 99% confidence interval for µ.
Margin of error for the 99% CI for μ: 260
1200
3500
)576.2(* ===
n
zm
σ
99% CI for μ: )3433,2913(2603173 =±=± mx
The larger the value of C, the wider the interval.

18. 18
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
Standarddeviationσ⁄√n

19. 19
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
*
* 





=⇔=
m
z
n
n
zm
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.
209254.2091
150
3500*96.1*
22
≈=





=





=
m
z
n
σ
Would we need a much larger sample size to obtain a margin of
error of $100?
Choosing the Sample Size