Statistics Chapter 7 (Bentley)

Control Chart

Most commonly used statistical tool in quality control A plot of calculated statistics of the production process over time. If the calculated statistics fall in an expected range, then the production process is in control. If the calculated statistics reveal an undesirable trend, then adjustment of the production process is likely necessary.

Clusters

Mutually exclusive and collectively exhaustive groups for cluster sampling.

Strata

Mutually exclusive and collectively exhaustive groups for stratified random sampling.

Sample Statistics (or "Statistic")

Used to make inferences about the unknown population parameter

Statistical Quality Control

Involves statistical techniques used to develop and maintain a firm's ability to produce high-quality goods and services.

Parameter

Is a constant, although its value may be unknown.

Statistics

Such as the sample mean or the sample proportion, is a random variable whose value depends on the chosen random sample.

Classic Case of a "Bad" Sample

The Literary Digest Debacle of 1936 *Note more information on pages 212-213

Estimator

When a statistic is used to estimate a parameter, it is referred to as an _____.

Estimate

A particular value of the estimator

Assignable Variation

Caused by specific events or factors that can usually be identified and eliminated. Firm wants to identify and correct these types of variations in the production process.

Point Estimator

Common to refer to the estimator as this because it provides a single value - a point - as an estimate of the unknown population parameter.

Population

Consists of all items of interest in a statistical problem

Expected Value of the Sample Mean X (with line over it)

Equals the population mean, or E(X (with line over it))=μ. In other words, the sample mean is an unbiased estimator of the population mean.

Upper Control limit (UCL) Equation

Expected Value + (3*Standard Error)

Lower Control Limit (LCL) Equation

Expected Value - (3*Standard Error0

The Finite Population Correction Factor for the Sample Mean

Used to reduce the sampling variation of X (with a line over it). The resulting standard error is se(X(with a line over it))=σ/(sqr(n))*(sqr((N-n)/N-1))

The Finite Population Correction Factor for the Sample Proportion

Used to reduce the sampling variation of the sample proportion P (with a line over it). The resulting standard error of P (with a line over it) is se(P (with a line over it))=sqr((p(1-p))/n)*sqr((N-n)/(N-1)). The transformation of P (with a line over it) to Z is made accordingly.

Variance of X (with line over it)

Var(X (with line over it))=(σ^2)/sqr(n)

Point Estimate

Another name for an estimate

Simple Random Sample

Basic type of sample that can be used to draw statistically sound conclusions about a population as any other sample of n observations. Most statistical methods presume simple random samples.

Sampling from a Normal Population

For any sample size n, the sampling distribution of σ is normal if the population X from which the sample is drawn is normal distributed. If X (with line over it) is normal, we can transform it into a standard normal random variable as: Z=(X (with line over it)-E(X (with line over it)))/se(X (with line over it))=(X (with line over it)-μ)/(σ/sqr(n). Therefore, any value x (with line over it) on X (with line over it) has a corresponding value z on Z given by z =(x(with line over it) - μ)/(σ/dqr(n))

Standard Error of the Sample Mean

Standard deviation of the sample mean X (with line over it) Equals the population standard deviation divided by the sqr of the sample size, that is, se(X (with line over it))=σ/(sqr(n))

Qualitative Data Control Charts

-the p (with a line over it), which monitors the proportion of defectives (or some other characteristics) in a production process -the c chart, which monitors the count of defects per item, such as the number of blemishes on a sampled piece of furniture

Quantitative Data Control Charts

-x (with a line over it) chart, which monitors the central tendency of a production process -the R chart and the s chart, which monitor the variability of a production process

Control Chart Characteristics

1. A control chart plots the sample estimates, such as x (with a line above it) or p (with a line above it). So as more and more samples are taken, the resulting control chart provides one type of safeguard when assessing if the production process is operating within predetermined guidelines. 2. All sample estimates are plotted with reference to a centerline. The centerline represents the variables's expected value when the production process is in control. 3. In addition to the centerline, all control charts include an upper control limit and a lower control limit. These limits indicate excessive deviation above (upper control limit) or below (lower control limit) the expected value of the variable of interest. A control chart is valid only if the sampling distribution of the relevant estimator is (approximately) normal. Under the assumption, the control limits are generally set at three standard deviations from the centerline. As we observed in Chapter 6, the area under the normal curve that corresponds to +/-3 standard deviation from the expected value is 0.9973. Thus, there is only a 1-0.9973=0.0027 chance that the sample estimates will fall outside the limit boundaries.

Detection Approach

A preferred approach to quality control. A firm using this approach inspects the production process and determines at which point the production process does not conform to specifications. The goal is to determine whether the production process should be continued or adjusted before a large number of defects are produced.

Acceptance Sampling

An approach for statistical quality control (one of two common ones). Used it if produces a product (or offers a service) and at the completion of the production process, the firm then inspects a portion of the products. If a particular product does not conform to certain specifications, then it is either discarded or repaired. The problems with this approach to quality control are, first, it is costly to discard of repair a product. Second, the detection of all defective products is not guaranteed. Defective products may be delivered to customers, thus damaging the firm's reputation.

Chance Variation

Caused by a number of randomly occurring events that are part of the production process. Not generally considered under the control of the individual worker or machine. Is expected and not a source of alarm in the production process so long as its magnitude is tolerable and the end product meets acceptable specifications.

Central Limit Theorem for the Sample Proportion

For any population proportion p, the sampling distributing of P (with a line over it) is approximately normal if the sample size n is sufficiently large. As a general guideline, the normal distribution approximation is justified when np >= 5 and n(1-p)>=5. If P (with a line over it) is normal, we can transform it into the standard normal random variable as Z=(P(with a line over it)-E(P(with a line over it)))/se(P(with a line over it))=(P(with a line over it)-p)/sqr((p(1-p)/n). Therefore, any value p (with a line over it) on P(with a line over it) has a corresponding value z on Z given by z=(p(with a line over it)-p)/sqr(p(1-p)/n)

Stratified Random Sampling

In ____, the population is first divided up into mutually exclusive and collectively exhaustive groups, called strata. A stratified sample includes randomly selected observations from each stratum. The number of observations per stratum is proportional to the stratum's size in the population. The data for each stratum are eventually pooled.

Stratified Verses Cluster Sampling

In stratified sampling, the sample consists of observations from each group, whereas in cluster sampling, the sample consists of observations from the selected groups. Stratified sampling is preferred when the objective is to increase precision and cluster sampling is preferred when the objective is to reduce costs.

Central Limit Theorem (CLT)

Perhaps the most remarkable result of probability theory States the sum or the average of a large number of independent observations from the same underlying distribution has an approximate normal distribution. The approximation steadily improves as the number of observations increases. In other words, irrespective of whether or not the population X is normal, the sample mean X (with a line over it) computed from a random sample of size n will be approximately normally distributed as long as n is sufficiently large. For any population X with expected value μ and standard deviation σ, the sampling distribution of X (with line over it) will be approximately normal if the sample size n is sufficiently large. As a general guideline, the normal distribution approximation is justified when n>=30. As before, if X (with a line over it) is approximately normal, then we can transform it to Z = (X (with a line over it) - μ)/(σ/sqr(n))

Sampling Distribution of X (with line over it)

Probability distribution of the sample mean X (with line over it)

Nonresponsive Bias

Refers to a systematic difference in preferences between respondents and nonrespondents to a survey or a poll.

Selection Bias

Refers to the systematic underrepresentation of certain groups from consideration for the sample.

Bias

Refers to the tendency of a sample statistics to systematically over- or under- estimate a population parameter. It is often caused by samples that are not representative of the population.

Sample

Subset of the population Good if representative of the population attempting to be described

Expected Value and Standard Error of the Sample Proportion

The expected value of P (with a line over it) equals E(P(with a line over it)=p The standard deviation fo P (with a line over it) equals se(P(with a line over it))=sqr((p(1-p))/n)

Cluster Sampling

The population is divided up into mutually exclusive and collectively exhaustive groups, called clusters. A cluster sample includes the observations from randomly selected clusters.