Shewhart X bar control charts with estimated control limits are widely used in practice. When the sample size is not fixed,we propose seven statistics to estimate the standard deviation sigma . These estimators are applied to estimate the control limits of Shewhart X bar control chart. The estimated results through simulated computation are given and discussed. Finally, we investigate the performance of the Shewhart X bar control charts based on the seven estimators of sigma via its simulated average run length (ARL).

Statistical process control (SPC) procedures suitable for autocorrelated processes have been extensively investigated in recent years. The most popular method is the residual-based control chart. To implement this method, a time series model, which is usually an autoregressive moving average (ARMA) model, of the process is required. However, the model must be estimated from data in practice and the resulting ARMA modeling errors are unavoidable. Residual-based control charts are known to be sensitive to ARMA modeling errors and often suffer from inflated false alarm rates. As an alternative, control charts can be applied directly to the autocorrelated data with widened control limits. The widened amount is determined by the autocorrelation function of the process. The alternative method, however, can not be also free from the effects of modeling errors because it relies on an accurate process model to be effective. To compare robustness to the ARMA modeling errors between the preceding two kinds of methods for control charting autocorrelated data, this dissertation investigates the sensitivity analytically. Then, two robust design procedures for residual-based control charts are developed from the result of the sensitivity analysis. The first approach for robust design uses the worst-case (maximum) variance of a chart statistic to guarantee the initial specification of control charts. The second robust design method uses the expected variance of the chart statistic. The resulting control limits are widened by an amount that depends on the variance of chart statistic - maximum or expected - as a function of (among other things) the parameter estimation error covariances.

No description available.

In statistical process control, it is usually assumed that observations on the process output at different times are lID. However, for many processes the observations are correlated and control charts for monitoring these processes have recently received much attention. For monitoring the process level, this study evaluates the properties of control charts, such as the EWMA chart and the CUSUM chart, based on the residuals from the forecast values of an ARMA model. It is assumed that the process mean is a ftrst order autoregressive (AR(l)) model and the observations are the mean plus a random error. Properties of these charts are evaluated using a Markov chain approach or an integral equation approach. The performance of control charts based on the residuals is compared to the performance of control charts based on the original observations. A combined chart using forecasts and residuals as the control statistics as well as a combined chart using the EWMA of observations and the EWMA of residuals as the control statistics are also studied by simulation. It is found that no universally "good" chart exists among all the charts investigated in this study. In addition, for monitoring the process variance, two kinds of EWMA chart based on residuals are studied and compared. / Ph. D.

No description available.

Injury risk reduction is an ever-evolving topic within an athletic environment. Consequences from an injury include participation time loss, financial, social, and personal costs. Coaching and medical staff strive to reduce the risk through various manners. Training load monitoring is one method that is utilized in injury risk reduction through global positioning systems (GPS) with statistical modeling. The purpose of this study was to investigate the external loads for training sessions and competition in starters versus non-starters; to determine if there were control chart violations associated with sustained injuries; and to determine whether in-season injuries were associate with one or more control chart violations. NCAA Division I female soccer players were recruited during the fall 2019 season. Participants were provided a STATSports GPS unit to wear during all practice and competition sessions to analyze the following variables: total distance, high metabolic load distance, sprints, accelerations, decelerations, and dynamic stress load (DSL). These variables were analyzed using statistical process control charts (SPC Charts) and Nelson Rules. Overall, there were 1,235 violations for the team, with the highest amount coming from DSL. Throughout the season, there were 16 time-loss injuries. Within the 3- and 7-day periods prior to injury, there were only two cases in which the injured athlete had more violations when compared to the team average. Therefore, SPC Charts were not a good indicator of injury risk prediction within this population. Future research includes reassessing these methods within a larger population and for a longer duration (i.e. several seasons). / Master of Science / Reducing the risk of injury in athletes is a focal point for many coaches, training, and medical staffs in collegiate athletics. The consequences of injury range from loss of playing time to financial and long-term health costs. Being able to reduce the risk of injuries not only has personal implications for the athlete but also relates to overall team success. Using global positioning systems (GPS) to track the amount of work done in training can possibly reduce injury risk. This study planned to investigate the workload in NCAA Division 1 collegiate female soccer athletes and if any injuries were sustained during both training and competition settings. The results suggest that statistical process control (SPC) charts and the Nelson Rules did not predict injury risk within this population. There is limited research that has used these tools. Future work can reassess these methods within larger collegiate athletic populations, over a longer period of time.

A control chart that simultaneously tracks the mean and variance of a normally distributed variable with no compensation effect is defined in this work. This joint control chart is compared to five other charts: an Χ chart, an s² chart, a Reynolds and Ghosh chart, a Repko process capability plot, and a t-statistic chart. The criterion for comparison is the probability of a Type II sampling error. Several out-of-control cases are examined. In the case of Repko, an equation is defined to compute the Type II error probability. The results indicate that the Reynolds and Ghosh statistic is powerful for cases when the variance shifts out of control. The Χ chart is powerful when the mean shifts with moderate changes in the variance. The joint chart is powerful for moderate changes in the mean and variance. / Master of Science

Most control charts monitor only one quality characteristic. There are, however, many manufactured products for which good quality requires meeting specifications in more than one physical characteristic. Typical practice when dealing with multiple quality characteristics is to take a separate sample for each characteristic and then create individual univariate control charts which are independently monitored. This method can result in errors due to not accounting for the effects of correlation. In order to avoid these errors, an alternate approach to multivariate quality control problems is proposed and studied here. The original problem is converted into a univariate problem by using the following transformation: y=Σ a_ix_i i where αi = weighting coefficient for the i^th quality characteristic X_i = represents the i^th quality characteristic This transformation retains sensitivity to changes in the original quality variables. The resulting univariate quality control model takes into account the sampling error probabilities for each of several candidate hypotheses. The probabilities of correctly diagnosing process shifts when an out-of-control state occurs are calculated and tabulated as are the probabilities that the model will signal when an out-of-control state occurs. / Master of Science

In recent years, variable sampling interval (VSI) control charts have been intensively investigated. In contrast to traditional fixed sampling interval (FSI) control charts, VSI charts vary the sampling interval as a function of the data. VSI charts detect many process changes faster than their FSI counterparts. A disadvantage, however, of VSI charts as recently formulated is that the advance prediction of sampling times is impossible for more than the next sample. A control chart is proposed which applies a sequential probability ratio test (SPRT) at fixed sampling intervals, the SPRT chart, to monitor the mean of a process with a normal distribution. A natural modification of the SPRT chart, the SPRT chart with sampling at fired times (SFT), is also proposed in which samples are always taken at pre-specified, equally spaced fixed times, with additional samples taken between these times as indicated by the data. A third control chart is introduced as a generalization of the VSI cumulative sum (CUSUM) chart that uses two sampling intervals, called the universal CUSUM (UC) chart, in order to address the need for a general framework for the study of control charts that are equivalent to a sequence of SPRT’s. The UC chart can also be viewed as a generalization of the SPRT chart. The integral equation approach is adapted for the evaluation of properties of both the unmodified and modified with SFT versions of the SPRT chart, such as average time to signal (ATS), steady state ATS (SSATS), and average number of observations to signal (ANOS). After comparisons are performed within the general framework of the UC chart, the unmodified SPRT chart is found to be more efficient than both the FSI and VSI X charts and the FSI CUSUM chart, though very similar in efficiency to the VSI CUSUM chart. The modified SPRT chart with SFT is found to be more efficient than all five of the other control charts, including its unmodified version and the VSI CUSUM chart. General guidelines are provided for the design of both versions of the SPRT chart. / Ph. D.

The standard Shewhart control chart for monitoring process stability is generalized by selecting a point in time at which the distance between the control limits is reduced. Three cost models are developed to describe the total cost per unit time of monitoring the mean of a process using both the standard and the generalized Shewhart control chart. The cost models are developed under the assumption that the quality characteristic of interest is normally distributed with known and constant variance. In the development of the first model, the negative exponential distribution is employed to model the time to process shift. Then, the uniform distribution and the Weibull distribution are used for the same purpose in the second and the third model, respectively. The motivation for this effort is to increase chart sensitivity to small but anticipated shifts in the process average. Cost models are constructed to allow the optimal choice of change over time and the best values for the initial and adjusted control limit values. The cost models are analyzed to determine the optimal control chart parameters including those associated with both the standard and the generalized control chart. The models are also used to provide a comparison with conventional implementation of the control chart. It is shown that the proposed cost models are efficient and economical. Figures and tables are provided to aid in the design of models for both the standard and the generalized Shewhart control chart. / Ph. D.