chapter 7 stat

Beer bottles are filled so that they contain an average of 450 ml of beer in each bottle. Suppose that the amount of beer in a bottle is normally distributed with a standard deviation of 8 ml. What is the probability that a randomly selected bottle will have less than 444 ml of beer?

(x-μ) / (444-450)/8 =-.75 P(x<444) = P(z<-.75) find corresponding z value = .2266

Is the sampling distribution of the sample proportion approximately normal with n = 11 and n = 72?

11 no 72 yes

Calculate the standard error for the sampling distribution of the sample proportion when n = 11

= sqrt (.36(1-.36))/11

standard error w proportions =

= sqrt (p(1-p) / n)

standard error =

= σ / sqrt n

Control Charts

A plot of calculated statistics of the production process overtime production process is in control if it falls in an expected range production process is our of control if calculated statistics reveal an undesirable trend qualitative data is x chart qualitative data is p chart

Acceptance Sampling

Used at the completion of a production process or service If a particular product does not conform to certain specifications, then it is either discarded or repaired it is costly to discard or repair a product detention of all defective products is not garanteed

Assignable Variation

caused by specific events or factors that can usually be identified and eliminated identified and corrected

population

consist of all the items of interest in a statistical problem

Cluster Sampling

divide population into mutually exclusive and collectively exhaustive groups called clusters

stratified random sampling

divide the population into mutually exclusive and collectively exhaustive groups, called strata

advantages and disadvantages of cluster sampling

less expensive than other sampling methods less precision than other sampling methods *useful when clusters occur naturally in the population (city blocks, schools, and the geographic areas)

p chart

monitors the proprotion of defectives relies on central limit theorem for normal approximation for the sampling distribution of the sampling proportion centerline - the expected proportion when the process is under control UCL = centerline + 3 * se from centerline LCL = centerline - 3 * se from centerline p + 3 (sqrt p (1 - p) / n)

strata

mutually exclusive and collectively exhaustive groups strata are based on one or more classification criteria

central limit theorem for proportion

np≥5 and n(1-p)≥5

stratum

number of observations per stratum is proportional the stratum's size in the population

Population is described by

parameters a parameter is a constant, whose value may be unknown

most statistical methods

presume simple random samples

x chart

qualitative data centerline - the mean when the process is under control upper control limit (UCL) = centerline + 3 * se mean lower control limit (LCL) = centerline - 3 * se μ + 3(σ/ sqrt n)

randomly select cluster

sample every observation in those randomly selected clusters

Sample is described by

statistics a statistic is a random variable whose value depends on the chosen random sample used to make inferences about the population parameters

Stratified vs Cluster

stratified sampling consists of elements from each group preferred when the objective is to increase precision Cluster sampling consists of elements from the selected groups preferred when the objective is to reduce costs

Sample

subset of a the population

Sample distribution of the mean

the frequency or probability distribution of sample means derived from all samples of a given size n

Bias

the tendency of a sample statistic to systematically over or underestimate a population parameter

Finite Population Correction Factor for Proportions

used to reduce the sampling of p bar applies when the sample size is large relative to the population size ( n ≥ 0.05N) se (Pbar) = sqrt (p(1-p) / n) * (sqrt N-n / N-1) transformation of x to z Is made accordingly

The Finite Population Correction Factor

used to reduce the sampling variation of Xbar applies when the sample size is large relative to the population size ( n ≥ 0.05N) ex- 470 * .05 = 23.5 38 > 23.5 it applies Expected Value E(Xbar) = μ Standard Error of Xbar = σ/sqrt n * (sqrt (N-n)/(N-1) transformation of x to z is made accordingly

**Calculate the probability that the sample proportion is between 0.34 and 0.36 for n = 72.

z = (p bar - p) / sqrt (p( 1 - p) / n) z = (.34 - .36) / sqrt (.36 (1 - .36)/72)) = -.3536 z = (.36 -.36) / sqrt (.36(1-.36) / 72) = 0 -.35 ≤ xbar ≤ 0 p(z ≤ 0) - p(z ≤ -.35) .5 - .3632 = .1368

Beer bottles are filled so that they contain an average of 450 ml of beer in each bottle. Suppose that the amount of beer in a bottle is normally distributed with a standard deviation of 8 ml. What is the probability that a randomly selected 6-pack of beer will have a mean amount less than 444 ml?

z = (xbar - μ) / (σ /sqrt n) z = (444 - 450) / (8/ sqrt 6) = -1.84 P (xbar < 444) = P (z < 1.84) find corresponding z value = .0329

Sampling distribution of X bar Is the sampling distribution of the sample mean with n = 17 and n = 43 normally distributed

No, only the sample mean with n = 43 will have a normal distribution bc central limit theorem 43> 30

Estimate

a particular value of the estimator value of that function taken on a particular sample

Estimator

a statistic that is used to estimate a population parameter a function that maps samples into your parameter space

nonresponse bias

a systematic difference in preferences between respondents and non respondents to a survey a poll

selection bias

a systematic underrepresentation of certain groups from consideration for the sample

advantages

all population subdivisions are represented in the sample parameter estimates have greater precision than those estimated from simple random sampling

**Calculate the probability that the sample mean falls between 70 and 72 for n = 43. (only can do for normally distributed) formula for sampling distribution

any value of x bar of X bar has a corresponding value z of Z given by z = (xbar - μ) / (σ /sqrt n) find z = (70-70) / (5.8/sqrt 43) = 0 find z = (72-70) / (5.8/sqrt 43) =2.26 70≤ xbar ≤ 72 = 0≤ xbar≤ 2.26 = P(z ≤ 2.26) - (P(z≤0) *z table .9881 - .5 = .4881

chance variation

caused by a number of randomly occurring events that are part of the production process not controllable by the individual worker or machine expected, so not a source of alarm as long as its magnitude is tolerable and the end product meets specifications

Detection Approach

inspection occurs during the production process in order to detect any nonconformance to specifications goal is to determine whether the production process should be continued or adjusted before producing a large number of defects types of variations: chance variation assignable variation

Statistical Quality Control

involves statistical techniques used to develop and maintain a firm's ability to produce high quality goods and services - acceptance sampling - detection approach

simple random sample

is a sample of n observations that has the same probability of being selected from the population as any other sample of n observations

sample statistic

is calculated from sample and used to make inferences about the population parameter

population parameter

is unknown