BCOR 1020Business Statistics

Lecture 18 – March 20, 2008

Overview

• Chapter 8 – Sampling Distributions and Estimation– Confidence Intervals

• Binomial proportion ()

– Sample size determination for a proportion ()

Chapter 8 – Confidence Interval for a Binomial Proportion ()

Estimating a binomial proportion:

n

Xp ̂

• Recall, an unbiased estimate of in Bernoulli trials is…

where X is the number of successes observed in n trials.

• We can determine the mean (expected value) and standard error (standard deviation) of this estimator…

p = (1-)n

p

Since is unknown, we often estimate this standard error with

npp

p)1(

Chapter 8 – Confidence Interval for a Binomial Proportion ()

The Central Limit Theorem and the Sampling Distribution of p:

n

X

• Also recall, that a Binomial random variable is approximately normal if n is large.

• Generally, if n > 10 and n(1 – ) > 10.

• So, if n is sufficiently large, then our unbiased estimate of in these Bernoulli trials is also approximately normal.

• p = is approximately normal with

p npp

p)1( and

Chapter 8 – Confidence Interval for a Binomial Proportion ()

The Central Limit Theorem and the Sampling Distribution of p:

• Applying the standard normal transformation to p leads us to the conclusion that…

npp

pZ

)1(

is approximately a standard normal random variable!

• Just as before, we can choose a value of the standard normal, z, such that P(-z < Z < z) = 100(1-)% {95% for example} and use this to derive a confidence interval…

Chapter 8 – Confidence Interval for a Binomial Proportion ()

npp

nppp zpzzz

npp

)1()1()1(

npp

npp zpzp )1()1(

npp

npp zpzp )1()1(

)1,0(~)1(N

pZ

npp

Since P(-z < Z < z) = 100(1-)% and

Is the 100(1 – )% Confidence Interval for .

Chapter 8 – Confidence Interval for a Binomial Proportion ()

• As n increases, the statistic p = x/n more closely resembles a continuous random variable.

• As n increases, the distribution becomes more symmetric and bell shaped.

• As n increases, the range of the sample proportion p = x/n narrows.

• The sampling variation can be reduced by increasing the sample size n.

Applying the CLT:

Chapter 8 – Confidence Interval for a Binomial Proportion ()

• A sample of 75 retail in-store purchases showed that 24 were paid in cash. What is p?

Example Auditing:

p = x/n = 24/75 = .32

• Is p normally distributed?

np = (75)(.32) = 24

n(1-p) = (75)(.88) = 51

Both are > 10, so we may conclude normality.

Chapter 8 – Confidence Interval for a Binomial Proportion ()

Example Auditing:

• The 95% confidence interval for the proportion of retail in-store purchases that are paid in cash is:

p(1-p)n

p + z = .32(1-.32)

.32 + 1.96

= .32 + .106

.214 < < .426

• We are 95% confident that this interval contains the true population proportion.

Clickers

Suppose your business is planning on bringing a new product to market. At least 20% of your target market is willing to pay $25 per unit to purchase this product. If 200 people are surveyed and 44 say they would pay $25 to purchase this product, estimate .

(A) p = 0.11

(B) p = 0.22

(C) p = 0.44

(D) p = 0.50

Clickers

If 200 people are surveyed and 44 say they would pay $25 to purchase this product, is the sample large enough to treat p as normal?

(A) Yes

(B) No

ClickersFind the z-value for a 95% C.I. on the binomialproportion,

(A) = 1.282 (B) = 1.645

(C) = 1.960 (D) = 2.576

Clickers

Suppose your business is planning on bringing a new product to market. At least 20% of your target market is willing to pay $25 per unit to purchase this product. If 200 people are surveyed and 44 say they would pay $25 to purchase this product, find the 95% confidence interval for .

(A)

(B)

(C)

(D) 220.022.0 110.022.0

029.022.0 057.022.0

Chapter 8 – Sample Size Determination for a C. I. on

• The width of the confidence interval for p depends on- the sample size- the confidence level- the sample proportion p

• To obtain a narrower interval (i.e., more precision) either- increase the sample size or- reduce the confidence level

Narrowing the Interval:

Chapter 8 – Sample Size Determination for a C. I. on

• To estimate a population proportion with a precision of + E (allowable error), you would need a sample of size

• Since is a number between 0 and 1, the allowable error E is also between 0 and 1.

Sample Size Determination for a Proportion:

• Since is unknown, we will either use • a prior estimate, p.• or the most conservative estimate = ½.

12

Ezn

Chapter 8 – Sample Size Determination for a C. I. on

• A sample of 75 retail in-store purchases showed that 24 were paid in cash. We calculated the 95% confidence interval for the proportion of retail in-store purchases that are paid in cash:

38516.384)5)(.5(.)1( 2

05.096.12 E

zn

Example Auditing:

= .32 + .106 .214 < < .426or

• If we want to calculate a confidence interval that is no wider than + 0.05, how large should our sample be?

• Using our estimate, p = 0.32,

• Using the most conservative estimate, p = 0.5,

33537.334)68)(.32(.)1( 2

05.096.12 E

zn

Clickers

Suppose your business is planning on bringing a new product to market. At least 20% of your target market is willing to pay $25 per unit to purchase this product. Using our earlier estimate, p = 0.22, determine how large our sample should be if we want a confidence interval no wider than E = + 0.05.

(A) n = 40

(B) n = 264

(C) n = 338

(D) n = 1537

A Look Ahead to Chapter 9

• Chapter 9 – Hypothesis Testing– Logic of Hypothesis Testing

Chapter 9 – Logic of Hypothesis Testing

What is a statistical test of a hypothesis?– Hypotheses are a pair of mutually exclusive, collectively

exhaustive statements about the world. – One statement or the other must be true, but they cannot

both be true.– We make a statement (hypothesis) about some

parameter of interest.– This statement may be true or false.– We use an appropriate statistic to test our hypothesis.– Based on the sampling distribution of our statistic, we can

determine the error associated with our conclusion.

Chapter 9 – Logic of Hypothesis Testing

5 Components of a Hypothesis Test:1. Level of Significance, – maximum probability of a

Type I Error • Usually 5%

2. Null Hypothesis, H0 – Statement about the value of the parameter being tested

• Always in a form that includes an equality

3. Alternative Hypothesis, H1 – Statement about the possible range of values of the parameter if H0 is false

• Usually the conclusion we are trying to reach (we will discuss.)• Always in the form of a strict inequality

Chapter 9 – Logic of Hypothesis Testing

5 Components of a Hypothesis Test:4. Test Statistic and the Sampling Distribution of the Test

statistic under the assumption that H0 is true• Z and T statistics for now

5. Decision Criteria – do we reject H0 or do we “accept” H0?

• P-value of the test

or• Comparing the Test statistic to critical regions of its distribution

under H0.

Chapter 9 – Logic of Hypothesis Testing

Error in a Hypothesis Test:

• Type I error: Rejecting the null hypothesis when it is true.

= P(Type I Error) = P(Reject H0 | H0 is True)

• Type II error: Failure to reject the null hypothesis when it is false.

= P(Type II Error) = P(Fail to Reject H0 | H0 is False)