Lecture 8: More on the Binomial Distribution and Sampling Distributions June 1, 2004 STAT 111 Introductory Statistics

Today’s Topics More on the binomial distribution –Mean and variance Sample proportion Normal approximation of the binomial Continuity correction Sampling distribution of sample means Central Limit Theorem

Recall: The Binomial Setting There are a fixed number n of trials. The n trials are all independent. Each trial has one of two possible outcomes, labeled “success” and “failure.” The probability of success, p, remains the same for each trial.

Recall: The Binomial Distribution The distribution of the count X of successes in the binomial setting is called the binomial distribution with parameter n and p, where –n is the number of trials –p is the probability of a success on any trial The count X is a discrete random variable, typically abbreviated as X ~ B(n, p). Possible values of X are the whole numbers from 0 to n.

The Binomial Distribution If X ~ B(n,p), then Examples: Let n = 3.

Developing Binomial Probabilities for n = 3 S1S1 F1F1 S2S2 F2F2 S2S2 F2F2 F3F3 S3S3 F3F3 S3S3 F3F3 F3F3 S3S3 S3S3 p 1-p p p p p p p P(SSS) = p 3 P(SSF) = p 2 (1 – p) P(SFS) = p 2 (1 – p) P(SFF) = p(1 – p) 2 P(FSS) = p 2 (1 – p) P(FSF) = p(1 – p) 2 P(FFS) = p(1 – p) 2 P(FFF) = (1 – p) 3

P(X = 0) = (1 – p) 3 P(X = 1) = 3p(1 – p) 2 P(X = 2) = 3p 2 (1 – p) P(X = 3) = p 3 Let X be the number of successes in three trials. X=0 X=1 X=2 X=3 Binomial Probabilities for n = 3 P(FFF) = (1 – p) 3 P(SSF) = p 2 (1 – p) P(SFS) = p 2 (1 – p) P(SFF) = p(1 – p) 2 P(FSS) = p 2 (1 – p) P(FSF) = p(1 – p) 2 P(FFS) = p(1 – p) 2 P(SSS) = p 3

Example: Rolling a Die Roll a die 4 times, let X be the number of times the number 5 appears. “Success” = get a roll of 5, so P(Success) = 1/6. X = 0 X = 1 X = 2 X = 3 X = 4

Example: Rolling a Die Find the probability that we get at least 2 rolls of 5.

Expected Value and Variance of a Binomial Random Variable If X~B(n,p),then

Let X i indicate whether the i th trial is a success or failure by, X 1, …, X n are independent and identically distributed with probability distribution Set-up for Derivation Outcome:10 Probability:p1-p X i =1, if i th trial is a success i = 1,2,….,n. X i =0, if i th trial is a failure.

Binomial Example: Checkout Lanes A grocery store has 10 checkout lanes. During a busy hour the probability that any given lane is occupied (has at least one customer) is Assume that the lanes are occupied or not occupied independently of each other. –What is the probability that a customer will find at least one lane unoccupied? –What is the expected number of occupied lanes? –What is the standard deviation of the number of occupied lanes?

Sample Proportions In statistical sampling we often want to estimate the proportion p of “successes” in a population. The sample proportion is defined as If the count X is B(n, p), then the mean and standard deviation of the sample proportion are

Sample Proportions Our sample proportion is an unbiased estimator of the population proportion p. The variability of our estimator decreases as sample size increases. In particular, we must multiply the sample size by 4 if we want the cut the standard deviation in half.

Sample Proportions The histogram of the distribution of the sample proportion when n = 1000, p = 0.6

Normal Approximation for Counts, Proportions Let X be the number of successes in a SRS of size n from a large population having proportion p of successes, and let the sample proportion of successes be denoted by Then for large n, –X is approximately normal with mean np and variance np(1 – p). – is approximately normal with mean p and variance p(1 – p) / n.

Normal Approximation: Rule of Thumb The accuracy of the approximation generally improves as the sample size n increases. For any fixed sample size, the approximation is most accurate when p is close to 0.5, and least accurate when p is near 0 or 1. As a general rule of thumb, then, we use the normal approximation for values of n and p such that np ≥ 10 and n(1 – p) ≥ 10.

Example The Laurier Company’s brand has a market share of 30%. Suppose that in a survey, 1,000 consumers of the product are asked which brand they prefer. What is the probability that more than 32% of the respondents will say they prefer the Laurier brand?

Another Example A quality engineer selects an SRS of size 100 switches from a large shipment for detailed inspection. Unknown to the engineer, 10% of the switches in the shipment fail to meet the specifications. The actual binomial probability that no more than 9 of the switches in the sample fail inspection is P(X ≤ 9) = How accurate is the normal approximation for this probability?

Another Example (cont.) Let X be the number of bad switches; then X ~ B(100, 0.1). It’s not that accurate. Note that np = 10, so n and p are on the border of values for which we are willing to use the approximation.

Continuity Correction While the binomial distribution places probability exactly on X = 9 and X = 10, the normal distribution spreads probability continuously in that interval. The bar for X = 9 in a probability histogram goes from 8.5 to 9.5, but calculating P(X ≤ 9) using the normal approximation only includes the area to the left of the center of this bar. To improve the accuracy of our approximation, we should let X = 9 extend from 8.5 to 9.5, etc.

Continuity Correction Use continuity correction to approximate the binomial probability P(X=10) when n=100, p=0.1 Using the normal approximation to the binomial distribution, X is approximately distributed as N(10, 3).

Continuity Correction The exact binomial probability is P(X=10)= P(X binomial =10)= P(9.5<X normal< 10.5)=

Continuity Correction Q: what about continuity correction for P(X<8)?

Continuity Correction Q: what about continuity correction for P(X>14)?

Example Re-visited Using the continuity correction, the probability that no more than 9 of the switches in the sample fail inspection is

Example: Inspection of Switches Find the probability that at least 5 but at most 15 switches fail the inspection.

Sampling Distributions Counts and proportions are discrete random variables; used to describe categorical data. Statistics used to describe quantitative data are most often continuous random variables. Examples: sample mean, percentiles, standard deviation Sample means are among the most common statistics.

Sampling Distributions Regarding sample means, –They tend to be less variable than individual observations. –Their distribution tends to be more normal than that of individual observations. We’ll see why later.

Sampling Distributions of Sample Means Let be the mean of an SRS of size n from a population having mean µ and standard deviation σ. The mean and standard deviation of are Why?

Sampling Distributions of Sample Means The shape of the distribution of the sample mean depends on the shape of the population distribution itself. One special case: normal population distribution Because: any linear combination of independent normal random variables is normal distributed.

Example The foreman of a bottling plant has observed that the amount of soda pop in each “32-ounce” bottle is actually a normally distributed random variable, with a mean of 32.2 ounces and a standard deviation of.3 ounce. –If a customer buys one bottle, what is the probability that that bottle contains more than 32 ounces? –If that same customer instead buys a carton of 4 bottles, what is the probability that the mean of those 4 bottles is greater than 32 ounces?

Example The starting salaries of M.B.A.s at Wilfrid Laurier Univ.(WLU) are normally distributed with a mean of $62,000 and a standard deviation of $14,500. The starting salaries of M.B.A.s at the University of Western Ontario (UWO) are normally distributed with a mean of $60,000 and a standard deviation of $18,300. –A random sample of 50 WLU M.B.A.s and a random sample of 60 UWO M.B.A.s are selected –What is the probability that the sample mean of WLU graduates will exceed that of the UWO graduates?

Central Limit Theorem When the population distribution is normal, so is the sampling distribution of What about when the population distribution is non-normal? For large sample sizes, it turns out that the distribution of gets closer to a normal distribution. As long as the population has finite standard deviation, this will be true regardless of the actual shape of the population distribution

Central Limit Theorem Formally, draw an SRS of size n from any population with mean µ and finite standard deviation σ. As n approaches infinity (gets very large) This can hold even if the observations are not independent or identically distributed. This is why normal distributions are common models for observed data.