Chapters 6 & 7 Overview

Created by Erin Hodgess, Houston, Texas

Chapter 6Confidence Intervals

6-1 Overview

6-3 Estimating a Population Mean: s Known

6-4 Estimating a Population Mean: s Not Known

6-2 Estimating a Population Proportion

Inferential Statistics

1. Estimate the value of a population parameter (Confidence Intervals)

2. Test some claim about a population parameter (Hypothesis Testing).

Chapters 6 & 7 mark the start of inferential statistics (drawing conclusions about population using sample data).

Two major types of inferential statistics: Use sample data to:

Quantitative vs. Qualitative

Quantitative Data (means) Population Mean, μ Sample Mean,

Qualitative Data (proportions) Population Proportion, p:

Sample Proportion, :

xp

N

x

p ˆx

pn

Estimating a Population Parameter (μ or p)

An estimate of a population parameter can be either:

1. Point Estimate, or

2. Interval Estimate

ex. Estimate μ, the mean weight of an adult male. ex. Estimate p, the proportion of adults

who have health insurance.

Estimating a Population Parameter, μ or p

Point Estimate: a single value estimate.

For μ, the best point estimate is x.

μ ≈ x.

For p, the best point estimate is

p ≈

Question: How can we improve our estimate?

p

p

Estimating a Population Parameter

Interval Estimate (Confidence Interval): a range of values that is likely to contain the parameter.

1. Format: μ = x ± E or p = p ± E

2. Comes with a level of confidence (1-α) which is the probability that the interval actually contains the parameter.

Common Confidence Levels

Confidence Levels:

(1-α)

90% 95% 99%

1-α: 0.90 0.95 0.99

α: 0.10 0.05 0.01

Created by Erin Hodgess, Houston, Texas

Section 6-3 Estimating a Population

Mean, μ (s Known)

Estimating a Population Mean, μ = x ± E

Margin of Error, E: the maximum likely difference observed between sample mean x and population mean µ.

How to find E? Back to the Central Limit Theorem:For n ≥ 30 or x ~ N, x ~ N (μ, σ/√n)

E x

Confidence Levels (1-α) and Critical Values

Critical Values zα/2: z-scores with an

area of α/2 in the upper tail.

(1-α) α α/2 zα/2

90% or 0.90 0.10 0.05 1.645

95% or 0.95 0.05 0.025 1.96

99% or 0.99 0.01 0.005 2.575

Critical Values

1. We know from Section 5-6 that under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution, as in Figure 6-2.

2. Sample proportions have a relatively small chance (with probability denoted by ) of falling in one of the red tails of Figure 6-2.

3. Denoting the area of each shaded tail by /2, we see that there is a total probability of that a sample proportion will fall in either of the two red tails.

4. By the rule of complements (from Chapter 3), there is a probability of 1— that a sample proportion will fall within the inner region of Figure 6-2.

5. The z score separating the right-tail is commonly denoted by z /2, and is referred to as

a critical value because it is on the borderline separating sample proportions that are likely to occur from those that are unlikely to occur.

Critical Values

The Critical Value

Figure 6-2

z2

Notation for Critical Value

The critical value z/2 is the positive z value that is at the vertical boundary separating an area of /2

in the right tail of the standard normal distribution. (The value of –z/2 is at the vertical boundary for the area of /2 in the left tail). The subscript /2 is simply a reminder that the z score separates an area of /2 in the right tail of the standard normal distribution.

DefinitionCritical Value

A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur. The number z/2 is a critical value that is a z score with the property that it separates an area of /2 in the right tail of the standard normal distribution. (See Figure 6-2).

Finding z2 for 95% Degree

of Confidence

-z2z2

Critical Values

2 = 2.5% = .025 = 5%

z2 = 1.96

Use Table A-2 to find a z score of 1.96

= 0.05

Finding z2 for 95% Degree

of Confidence

Procedure for Constructing a (1-α)*100% Confidence Interval

for µ when s is known

Assumptions: a) x comes from a SRS b) σ is knownc) n ≥ 30 or x ~ N

1. μ = x ± E ; where E = zα/2.σ/√n

2. Write as an interval: x – E < µ < x + E

x – E < µ < x + E

(x – E, x + E)

Confidence Interval Notation

μ = x ± E

Interval Notation:

Rounding: Round statistics and confidence intervals to one more decimal place than used in original set of data.

Notes on Confidence Intervals

Interpretation:•“There’s a 95% chance that the interval actually contains µ…” or, • “95% of all such samples would generate an interval that contained µ”

Desired Properties• High Confidence• Small Margin of Error

To Decrease Margin of Error• Lower level of confidence• Increase sample size

Determining Sample Size Needed to Estimate a Population Mean

(z/2) s n =

E

2

First Choose: 1. Confidence

2. Margin of Error

Then, the minimum sample size needed to estimate µ is:

Notes on Sample Size n 1. n depends only on confidence level and E.

Population size is irrelevant.

2. Since the formula gives the minimum sample size needed to achieve the desired confidence and margin of error, always round up to next larger whole number.

3. Must know σ in advance. Either: a. Use σ from a previous study b. Do a preliminary sample n ≥ 30 and s ≈ σc. If the range is known, use the range rule to

estimate σ. (σ ≈ range/4)

Created by Erin Hodgess, Houston, Texas

Section 6-4 Estimating a Population

Mean: s Not Known

1) The sample is a simple random sample.

2) Either the sample is from a normally distributed population, or n > 30.

Use Student’s t distribution:

s Not KnownAssumptions

If the distribution of a population is essentially normal, then the distribution of

is a Student t Distribution, with n -1 degrees of freedom

t =x - µ

sn

Student t Distribution (with n -1 degrees of freedom)

Degrees of Freedom (df ) corresponds to the number of sample values

that can vary after certain restrictions have been imposed on all data values

df = n – 1in this section.

Definition

Important Properties of the Student t Distribution

1. The Student t distribution is different for different sample sizes (see Figure 6-5 for the cases n = 3 and n = 12).

2. The Student t distribution has the same general symmetric bell shape as the normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples.

3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).

4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a s= 1).

5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.

Student t Distributions for n = 3 and n = 12

Figure 6-5

Confidence Interval for the Estimating µ, σ unknown

Based on an Unknown s and a Simple Random Sample from a Normally Distributed (or roughly symmetric)

Population

where E = t/2 ns

x – E < µ < x + E

t/2 with n-1 degrees of freedom from

Table A-3.

Figure 6-6

Using the Normal and t Distribution

Created by Erin Hodgess, Houston, Texas

Section 6-2 Estimating a Population

Proportion, p

Qualitative Data and Proportions

Population Proportion, p: proportion of the population that has the attribute

Sample Proportion: proportion of the sample that has the attribute

Other notation:

q = 1 – p: proportion of the population that does

not have the attribute. q = 1 – p: proportion of the sample that does not

have the attribute.

p x N

p p̂ x N

Estimating a Population Proportion, p

Point Estimate: p ≈ p Confidence Interval: p = p ± E

Notation:1. p = p ± E

2. p – E < p < p + E

3. (p – E, p + E)

Confidence Interval for Population Proportion

p = p ± E

z

E =

nˆ ˆp q

where

(1-α) Confidence Interval for Population Proportion

2

ˆ ˆˆ ; where

pqp p E E z

n

Notes:

1. Use z-distribution provided )

2. Round proportions and confidence intervals to

significant digits

ˆ ˆ5 and 5np nq

Sample Size for Estimating Proportion p

When an estimate of p is known: ˆ

Formula 6-2ˆ( )2 p qn =

ˆE2

z

When no estimate of p is known:

Formula 6-3( )2 0.25n =

E2

z