# Datastor Case Study - Stats Probabilitiy and Binomials

Case Study for DataStor Background DataStor, a data storage device and media manufacturer, produces a compact hard drive called DS1000, which stores 1GB of data - **Datastor Case Study - Stats Probabilitiy and Binomials** introduction. Their primary customer is Four-D, a national reseller of the drives. Four-D has rejected four shipments of drives from DataStor in the past 20 days. DataStor wants to understand why their shipments are being rejected. DataStor operates three 8-hour shifts, five days a week. Each shift produces approximately 120 drives for a daily average total of 360 drives per day.

The company runs quality checks called PDQ tests on one of their drives every hour of production. The test takes up to 20 minutes. Their historical “in control” process is defined with a mean of 7. 0 and a standard deviation of . 3. Four-D performs their own PDQ tests on the drives that DataStor ships them. They sample ten drives at random. If any of the drives have a PDQ score of 6. 2 or below, then the entire shipment will be rejected. Penalties are assessed for each unacceptable shipment. DataStor wants to determine why their shipments are being rejected.

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To do this, they first want to look at their internal processes before approaching Four-D to see if there are problems/issues on their side. Case Study Questions 1. If the DataStor DS1000 hard drive production process at DataStor Company is “in control”, what percentage of the drives produced would be considered to be in nonconformance by Four-D? In other words, what is the likelihood (probability) that the PDQ test score of a drive tested at DataStor will fall below 6. 2? The probability that a PDQ test score of a drive tested at DataStor will fall below 6. 2 is . 38%.

We arrived at this conclusion based on DataStor’s “in control” standard: Mean = 7. 0 Std Dev = . 3 Coincidentally, the mean and the X-bar of the data sample are the same, 7. 0 (6. 96). Using Normal Probabilities feature in PhStat, we determined: Common Data| Mean| 7| Standard Deviation| 0. 3| | | Probability for X <=| X Value| 6. 2| Z Value| -2. 666667| P(X<=6. 2)| 0. 0038304| 2. If the DataStor DS1000 hard drive production at DataStor Company is “in control”, how often will the shipments be found unacceptable by Four-D? That is, what is the probability of Four-D rejecting a shipment of drives from DataStor?

The probability of Four-D rejecting a shipment of drives from DataStor, if there process is “in control”, is 3. 8%. Four-D conducts the PDQ test on 10 random samples of each shipment.. We used this as the sample size in the Binomial Probability Distribution feature of PhStat. The probability of the event, that a drive would fall below Four-D’s quality standard of 6. 2, was gained from Question 1. Outcomes 1 – 10 were queried because there are 10 possible scenarios of Four-D rejecting the sample. The cumulative probability of Four-D rejecting a shipment from DataStor’s “in control” process is 3. 8%.

See the table below for the calculation. The binomial probability feature was used because this is a decision with two conditions: acceptable or unacceptable. Binomial Probabilities| | | | | | | Data| | | Sample size| 10| | | Probability of an event of interest| 0. 0038304| | | | | | | Statistics| | | Mean| 0. 0383038| | | Variance| 0. 0381571| | | Standard deviation| 0. 1953384| | | | | | | Binomial Probabilities Table| | | | X| P(X)| | | 1| 0. 037| | | 2| 0. 00064| | | 3| 6. 6E-06| | | 4| 4. 4E-08| | | 5| 2E-10| | | 6| 6. 5E-13| | | 7| 1. 4E-15| | | 8| 2. 1E-18| | | 9| 1. 8E-21| | | 10| 6. 8E-25| | | | | | | 0. 03765| | 3. What is the probability of DataStor having four or more shipments rejected in twenty days by Four-D, assuming that their production process has been “in control”? The probability of Four-D rejecting a 4 or more shipments in twenty days, based on DataStor’s “in control” process, is . 6%. To arrive at this probability, we used the PHStat Binominal Probability Distribution feature. This feature was selected because we’re looking for the probability of two conditions: acceptance or rejection of a shipment. The sample size is based on the 20 day sample. The probability of the event (. 3765) was determined in Question #2, the cumulative probability of Four-D rejecting a shipment (based on a sample of 10). The cumulative probability of outcomes from 4 to 20 is . 6%. The outcome parameters were 4 to 20 because, we were specifically asked to look at the probability of Four-D rejecting >=4 or more shipments in 20 days. Binomial Probabilities| | | | Data| | | Sample size| 20| | | Probability of an event of interest| 0. 03765| | | | | | | Statistics| | | Mean| 0. 753| | | Variance| 0. 72465| | | Standard deviation| 0. 851264| | | | | | | Binomial Probabilities Table| | | | X| P(X)| | | 4| 0. 005268| | | 5| 0. 00066| | | 6| 6. 45E-05| | | 7| 5. 05E-06| | | 8| 3. 21E-07| | | 9| 1. 67E-08| | | 10| 7. 2E-10| | | 11| 2. 56E-11| | | 12| 7. 52E-13| | | 13| 1. 81E-14| | | 14| 3. 54E-16| | | 15| 5. 54E-18| | | 16| 6. 77E-20| | | 17| 6. 24E-22| | | 18| 4. 07E-24| | | 19| 1. 67E-26| | | 20| 3. 28E-29| | | | 0. 005998| | 4. Based on your analysis in (1) – (3) above, what is your overall conclusion regarding the likelihood of DataStor having a frequent number of its recent shipments rejected by Four-D, if in fact their manufacturing process is “in control”?

Based on the data we have seen in Questions 1-3, there is probability of 0. 38 % finding a hard drive with PDQ score less than 6. 2. The probability of Four-D rejecting shipment is 3. 8%. The PDQ test sample data was provided for 3 shifts per day, for each of the 5 days of the week, for 10 weeks. We chose to divide the data according to the shifts to determine whether the quality of work for a given shift could explain the rejection rate from Four-D. The table below is based on the PDQ sample data. Shift 1 and 2 meet DataStor’s “in control” mean. Shift 3 does not meet the DataStor’s “in control” mean of 7. 0.

Similarly, Shifts 1 and are aligned with the sigma X-bar, based on their groups’ PDQ sample data. Shift 3’s sample standard deviation is higher than the sample sigma x-bar. | Mean| SD| Sigma X-bar| Shift 1| 7. 00| 0. 105| 0. 106| Shift 2| 7. 00| 0. 102| 0. 106| Shift 3| 6. 88| 0. 155| 0. 106| 5. What might be the source(s) of the problem at DataStor? Is the problem with rejected shipments due to an increase in drive nonconformance at DataStor, to increased quality requirements by Four-D Office Products, to damage during shipment, or is it simply due to random variation? What evidence leads you to your conclusion?

Following are the reasons for problems at DataStor: * The production of Shift 3 is not stable. The mean PDQ score is 6. 88 with a std deviation of 0. 155, this is higher than the sample std dev of the sample mean, which is 0. 106. Production for shifts 1 and 2 and within their target limits. Following graphs show that PDQ score in shift 3 falls below LCL * Datastor recently began plotting average of 8 PDQ score collected over each shift instead of individual values. This could be the reason why Datastor not able to identify the problem with quality. e. g. Lowest PDQ score for a shift is 6. 49.

As this is average of 8 PDQ score there is possibility that PDQ score may be lower than 6. 2. But Datastor is unable to identify it. The graphs below support questions 4 and 5. UCL/LCL| -3| -2| -1| X-bar| +1| +2| +3| Control Limit| 6. 68| 6. 79| 6. 89| 7. 00| 7. 11| 7. 21| 7. 32| Zones| -3? | -2? | -1? | Mean| 1? | 2? | 3? | | | | | | The chart shows Shift 1’s PDQ scores are within the upper and lower control limits. | | The chart shows Shift 2’s PDQ scores are within the upper and lower control limits. | | The chart shows Shift 3’s PDQ scores begin to fall outside of the lower controls during the 7th week. |