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BCOR 1020 Business Statistics. Lecture 18 – March 20, 2008. Overview. Chapter 8 – Sampling Distributions and Estimation Confidence Intervals Binomial proportion ( p ) Sample size determination for a proportion ( p ). p (1- p ) n. s p =.
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BCOR 1020Business Statistics Lecture 18 – March 20, 2008
Overview • Chapter 8 – Sampling Distributions and Estimation • Confidence Intervals • Binomial proportion (p) • Sample size determination for a proportion (p)
p(1-p)n sp = Chapter 8 – Confidence Interval for a Binomial Proportion (p) Estimating a binomial proportion: • Recall, an unbiased estimate of p 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… Since p is unknown, we often estimate this standard error with
Chapter 8 – Confidence Interval for a Binomial Proportion (p) The Central Limit Theorem and the Sampling Distribution of p: • Also recall, that a Binomial random variable is • approximately normal if n is large. • Generally, if np> 10 and n(1 – p) > 10. • So, if n is sufficiently large, then our unbiased estimate of • p in these Bernoulli trials is also approximately normal. • p = is approximately normal with and
Chapter 8 – Confidence Interval for a Binomial Proportion (p) The Central Limit Theorem and the Sampling Distribution of p: • Applying the standard normal transformation to p leads us to the conclusion that… is approximately a standard normal random variable! • Just as before, we can choose a value of the standard normal, za, such that P(-za < Z < za) = 100(1-a)% {95% for example} and use this to derive a confidence interval…
Chapter 8 – Confidence Interval for a Binomial Proportion (p) Since P(-za < Z < za) = 100(1-a)% and Is the 100(1 – a)% Confidence Interval for p.
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. Chapter 8 – Confidence Interval for a Binomial Proportion (p) Applying the CLT:
A sample of 75 retail in-store purchases showed that 24 were paid in cash. What is p? Chapter 8 – Confidence Interval for a Binomial Proportion (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.
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) 75 = .32 + 1.96 Chapter 8 – Confidence Interval for a Binomial Proportion (p) Example Auditing: = .32 + .106 .214 < p < .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 p. (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
Clickers Find the z-value for a 95% C.I. on the binomial proportion, p. (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 p. (A) (B) (C) (D)
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 Chapter 8 – Sample Size Determination for a C. I. on p Narrowing the Interval:
To estimate a population proportion with a precision of +E (allowable error), you would need a sample of size Chapter 8 – Sample Size Determination for a C. I. on p Sample Size Determination for a Proportion: • Since p is a number between 0 and 1, the allowable error E is also between 0 and 1. • Since p is unknown, we will either use • a prior estimate, p. • or the most conservative estimate p = ½.
Chapter 8 – Sample Size Determination for a C. I. on p Example Auditing: • 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: = .32 + .106 or .214 < p < .426 • 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,
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: • Level of Significance, a – maximum probability of a Type I Error • Usually 5% • Null Hypothesis, H0 – Statement about the value of the parameter being tested • Always in a form that includes an equality • 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: • Test Statistic and the Sampling Distribution of the Test statistic under the assumption that H0 is true • Z and T statistics for now • 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. • a = P(Type I Error) = P(Reject H0 | H0 is True) • Type II error: Failure to reject the null hypothesis when it is false. • b = P(Type II Error) = P(Fail to Reject H0 | H0 is False)