Chapter 7 Estimates, Confidence Intervals, and Sample Sizes Inferences Based on a Single Sample Estimation with Confidence Intervals

Chapter 7 Estimates, Confidence

Intervals, and Sample Sizes

Inferences Based on a Single Sample Estimation with Confidence Intervals

Learning Objectives

State what is estimated Distinguish point and interval estimates Explain interval estimates Estimating a population mean: is known. Estimating a population mean: is not known. Estimating a Population Proportion Compute Sample Size

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

EstimationHypothesis

Testing

Statistical Methods

Mean, , is unknown

Population Random Sample

Mean X= 79

Sample

I am 95% confident that is between

75 & 83.

Estimation Process

Unknown Population Parameters Are Estimated

XXMeanMean

ProportionProportion pp pp̂̂

VarianceVariance s

Estimate Population Parameter...

with the Sample Statistic

Jan. Feb. Mar. April May June July Aug. Sept. Oct. Nov. Dec.

2015 5.7 5.5 5.5 5.4 5.5 5.3 5.3 5.1 5.1

2014 6.6 6.7 6.7 6.3 6.3 6.1 6.2 6.1 5.9 5.8 5.8 5.6

2013 7.9 7.7 7.5 7.5 7.5 7.5 7.3 7.2 7.2 7.2 7.0 6.7

2012 8.3 8.3 8.2 8.1 8.2 8.2 8.2 8.1 7.8 7.9 7.8 7.8

2011 9.0 8.9 8.8 9.0 9.1 9.2 9.1 9.1 9.1 9.0 8.6 8.5

2010 9.7 9.7 9.7 9.9 9.7 9.5 9.5 9.6 9.6 9.6 9.8 9.4

2009 7.6 8.1 8.5 8.9 9.4 9.5 9.4 9.7 9.8 10.2 10.0 10.0

2008 4.9 4.8 5.1 5.0 5.5 5.6 5.8 6.2 6.2 6.6 6.8 7.2

National Unemployment Rates, 2008 - 2015 Source: Bureau of Labor Statistics

Point Estimation

PointEstimation

Interval

Estimation

Estimation

1. It provides a single value.

2. It is based on observations from 1 sample.

3. It gives no information about how close the value is to the unknown population parameter.

4. Example: Sample mean= 79 is a point estimate of the unknown population mean.

Point EstimationA point estimate of a parameter (p, or 22) is the value of a statistic ( , , or s

22) used to estimate the parameter.

p̂̂X

X

Interval Estimation

PointEstimation

Interval

Estimation

Estimation

The sample mean rarely equals the population mean. That is, sampling error is to be expected.

Therefore, in addition to the point estimate for , we need to provide some information that indicates the accuracy of the point estimate.

We do so by giving a confidence-interval estimate for .

Interval Estimation

A confidence interval or interval estimate is a range of values used to estimate the true value of a population parameter.

The interval is obtained from a point estimate of the parameter and a percentage that specifies the probability that the interval actually does contain the population parameter.

This percentage is called the confidence level of the interval.

Interval Estimation

Is the probability (when given in decimal form) that the confidence interval contains the unknown population parameter. The CL is denoted (1 - )

That isis the probability that the parameter is Not within the confidence interval.

Typical values of CL are 99%, 95%, 90%. This means that the corresponding values of are

0.01, 0.05, 0.10.

Confidence Level (CL)

The center of the interval is the point estimate E is called the margin of error and is half the

length of the confidence interval. E is determined by and the sample size n.

X .

Key Elements of the CI

This interval contains the parameter , ()% of the times.

Key Elements of the CI

( ) 1P X E X E

Margin of Error and CI

1. Provides Range of Values.

2. Is based on observations from 1 sample.

3. Gives information about how close the estimate is the to unknown population parameter.

2. Is stated in terms of probability.

3. Example: unknown population mean lies between 66 and 70 with 95% confidence.

Interval Estimation

http://onlinestatbook.com/stat_sim/sampling_dist/index.html

Margin of Error or Interval Width

Factors Affecting Interval Width

1. As data dispersion measured by increases the error or width E increases.

2. As Sample Size n increases the error or width E decreases.

3. As the level of confidence (1 - ) % increases the width increases because it affects /2Z

ConfidenceIntervals

Mean Variance Proportion

Known Unknown

Confidence Interval Estimates

CI for the Mean ( known)

1. Assumptions are Population standard deviation is known Population is normally distributed or Sampling distribution can be approximated by

normal distribution (n 30)

2. Confidence Interval Estimate

/ 2 / 2X Z X Zn n

Example 1

The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10.

0.05/ 2 0.05/ 2

10 1050 1.96 50 1.96

25 25 46.08 53.92

X Z X Zn n

You are a Quality Control inspector for Norton. The for 2-liter bottles is .05 liters. A random sample of 100 bottles showedX = 1.99 liters.

What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2

liter

2 litros

Tinto

Example 2

Solution

/2 /2

.05 .051.99 1.645 1.99 1.645

100 1001.982 1.998

X Z X Zn n


ConfidenceIntervals


Known Unknown

If X is a normally distributed variable with mean μ and standard deviation σ, then, for samples of size n, the variableXX is also normally distributed and has mean μ and standard deviation .

Equivalently, the standardized version ofXX ,

has the standard normal distribution.

/ n

/

Xz

n

CI for the Mean ( unknown)

In practice, σ is unknown therefore we cannot base our CI procedure on the standardized version ofXX.

The best we can do is estimate σ using the sample standard deviation s and replace σ with s in the equation

and base our CI procedure on the new variable,/

Xzn

/Xt

s n

CI for the Mean ( unknown)

t-Distributions and t-Curves

The t-distribution depends on n and for each n there is t-curve.

Notice, that to find a t-value you need to compute the sample mean and the sample standard deviation. This will usually require the use of a calculator.

Finding the t-Value Having a Specified Area to Its Right

For a t-curve with 13 degrees of freedom, find t0.05;

that is, find the t-value having area 0.05 to its right, as shown in the figure.

To find the t-value in question, we use Table IV. For ease of reference, we have repeated a portion of Table IV in the next slide.

Notice that the table provides the t-score only when you know the area to its right.

Unlike the table for z-scores, the t-table cannot be used to find probabilities when you know what the t-score is. For this, you need to use the t-distribution in the calculator.


For a t-curve with 13 degrees of freedom, find t0.05;

that is, find the t-value having area 0.05 to its right, as shown in the figure.


A random sample of n = 25 has X = 50 and s = 8. Set up a 95% confidence interval estimate for .

Example 1

/2, 1 /2, 1

8 850 2.0639 50 2.0639

25 2546.69 53.30

n n

S SX t X t

n n

You are a time study analyst in manufacturing. You have recorded the following task times (in minutes): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1.

What is the 90% confidence interval estimate of the population mean task time?

Example 2

0.05

3.7

0.38987

6 6 1 5

0.38987 / 6 0.1592

2.015

3.7 2.015 0.1592 3.7 2.015 0.1592

3.379 4.020

X

S

n df

S n

t

Solution

Finding Sample Sizes

Estimating the Sample Size

What sample size is needed to be 90% confident of being correct within 5? A pilot study suggested that the standard deviation is 45.

Example 1

2 2

/2 1.645 45219.2 220

5

Zn

E

You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that was about $400.

What sample size do you use?

Example 2

Solution2

0.025

21.96 400

50

245.86 246

Zn

E

ConfidenceIntervals


Known Unknown


Confidence Intervals for One Population

Proportion

Many statistical studies are concerned with obtaining the proportion (percentage) of a population that has a specified attribute.

For example, we might be interested in• the percentage of U.S. adults who have health insurance,

• the percentage of cars in the US that are imports,

• the percentage of U.S. adults who favor stricter clean air health standards, or

• the percentage of Canadian women in the labor force.

Proportion Notation and Terminology

Notice that in the previous examples, a given individual in the population will have the specified attribute or not.

This means that we are interested in an experiment that can have only two possible outcomes.

For instance,• A U.S. adult does have health insurance or does not.

• A car in the U.S. is either an import or is not.

• etcetera


We introduced some notation and terminology used when we make inferences about a population proportion.


Sometimes we refer to x (the number of members in the sample that have the specified attribute) as the number of successes and to n − x (the number of members in the sample that do not have the specified attribute) as the number of failures.


Notice that for a given sample of size n, the quotient x/n, is the mean number of successes in n trials. That is, is the mean of the variable X which is 1 when the member in the sample has the attribute and 0 when the member does not.


p̂̂

To make inferences about a population proportion p we need to know the sampling distribution of the sample proportion, that is, the distribution of the variable .

Because a proportion can always be regarded as a mean, we can use our knowledge of the sampling distribution of the sample mean to derive the sampling distribution of the sample proportion.

p̂̂

The Sampling Distribution of the Sample Proportion

The accuracy of the normal approximation depends on n and p. If p is close to 0.5, the approximation is quite accurate, even for moderate n. The farther p is from 0.5, the larger n must be for the approximation to be accurate.

As a rule of thumb, we use the normal approximation when np and n( − p) are both or greater.

In this section, when we say that n is large, we mean that np and n( − p) are both or greater.

Since in practice we do not know the value of p we replace the conditions np and n(− p) with the conditions and

This is the same as: the number of successes x and the number of failures n-x are both 5 or greater.

n(− p) ^̂np ^̂

CI for the Proportion p

Margin of Error or Interval Width

p̂

/ 2ˆ ˆ ˆ(1 ) /p z p p n / 2ˆ ˆ ˆ(1 ) /p z p p n

Estimating the Sample Size The margin of error E and CL (1-)% of a CI are

often specified in advance. We must then determine the sample size required to meet those specifications.

If we solve for n in the formula for E, we obtain

This formula cannot be used to obtain the required sample size because the sample proportion, , is not known prior to sampling.

2

/ 2ˆ ˆ(1 )z

n p pE

p̂̂

Estimating the Sample Size The way around this problem is to observe that the

largest can be is 0.25 when = 0.5p (-p)^̂ ^̂ p̂̂

Estimating the Sample Size Because the largest possible value of is 0.25

the most conservative approach for determining sample size is to use that value in equation

The sample size obtained then will generally be larger than necessary and the margin of error less than required. Nonetheless, this approach guarantees that the specifications will be met or bettered.

p (-p)^̂ ^̂

2

/ 2ˆ ˆ(1 )z

n p pE

Estimating the Sample Size

A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for proportion p of students that go to grad school.

Example 1

/2 /2

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ

.08 (1 .08) .08 (1 .08)0.08 1.96 0.08 1.96

400 400

0.053 0.107

p p p pp Z p p Z

n n

p

p

You are a production manager for a newspaper. You want to find the % defective newspapers. Of 200 newspapers, 35 had defects.

What is the 90% confidence interval estimate of the population proportion of defective newspapers?

Example 2

Solution

/2 /2

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ

.175 (.825) .175 (.825)0.175 1.645 0.175 1.645

200 200

0.1308 0.2192

p p p pp z p p z

n n

p

p

Example 3

Example 4

Example 5

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Chapter 7 Estimates, Confidence Intervals, and Sample Sizes Inferences Based on a Single Sample Estimation with Confidence Intervals