The quality engineering department suspects that both machines fill to the same net volume, whether or not this volume is An experiment is performed by taking a random sample from the output of each machine. Machine 1 Machine 2 The breaking strength of this plastic is important. The company will not adopt plastic 1 unless its breaking strength exceeds that of plastic 2 by at least 10 psi. Based on the sample information, should they use plastic 1?
Construct a 99 percent confidence interval on the true mean difference in breaking strength. The design engineers are interested in both the means and variance of the burning times. Type 1 Type 2 65 82 64 56 81 67 71 69 57 59 83 74 66 75 59 82 82 70 65 79 a Test the hypotheses that the two variances are equal. What is the P-value for this test? Check the assumption of normality for both types of flares. The assumption of normality is required in the theoretical development of the t-test.
However, moderate departure from normality has little impact on the performance of the t-test. The normality assumption is more important for the test on the equality of the two variances. An indication of nonnormality would be of concern here. The normal probability plots shown below indicate that burning time for both formulations follow the normal distribution. Yin and D. Jillie May, describes an experiment to determine the effect of C2F6 flow rate on the uniformity of the etch on a silicon wafer used in integrated circuit manufacturing.
No, C2F6 flow rate does not affect average etch uniformity. The box plots shown below indicate that there is little difference in uniformity at the two gas flow rates. Any observed difference is not statistically significant. See the t-test in part a. F :H :H ,,. Assume that the variances are equal. There is no evidence to indicate that the new filtering device has affected the mean Photoresist is a light-sensitive material applied to semiconductor wafers so that the circuit pattern can be imaged on to the wafer.
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After application, the coated wafers are baked to remove the solvent in the photoresist mixture and to harden the resist. Assume that all of the runs were made in random order. There appears to be a lower mean thickness at the higher temperature. This is also seen in the computer output. Provide a practical interpretation of this interval. This confidence interval doesnot include 0 in it, there for there is a difference in the two temperatures on the thickness of the photo resist. More samples are needed to detect a smaller difference. The time the part is allowed to cool in the mold before removal is thought to influence the occurrence of a particularly troublesome cosmetic defect, flow lines, in the finished housing.
After manufacturing, the housings are inspected visually and assigned a score between 1 and 10 based on their appearance, with 10 corresponding to a perfect part and 1 corresponding to a completely defective part. An experiment was conducted using two cool-down times, 10 seconds and 20 seconds, and 20 housings were evaluated at each level of cool- down time.
The data are shown below. From the computer output,.
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This interval does not contain 0. The two samples are different. The 20 second cooling time gives a cosmetically better housing. The data are as follows: 5. The normality assumption is much more important when analyzing variances then when analyzing means. A moderate departure from normality could cause problems with both statistical tests and confidence intervals. Specifically, it will cause the reported significance levels to be incorrect. The normal probability plot indicates that there is not any serious problem with the normality assumption.
P-Value: 0. As in any t-test, the assumption of normality is of only moderate importance. In the paired t-test, the assumption of normality applies to the distribution of the differences. That is, the individual sample measurements do not have to be normally distributed, only their difference.
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Do these plots support assumptions of normality and equal variance for both samples? The etch rate is an important characteristic of this process. Two different etching solutionsare being evaluated. See the Minitab output below. From the Minitab output, Specifically, tablet 1 is claimed to be absorbed twice as fast as tablet 2. Assume that and are known. How should we allocate the N observations between the two populations to obtain the most powerful test? Thus n1 and n2 are assigned proportionally to the ratio of the standard deviations.
This has intuitive appeal, as it allocates more observations to the population with the greatest variability. In general, describe how you created the data. Does this give you any insight regarding how the paired t-test works? A B delta 7. The fact that the difference between pairs is large makes the pooled estimate of the standard deviation large and the two-sample t-test statistic small. Therefore the fairly small difference between the means of the two treatments that is present when they are applied to the same experimental unit cannot be detected. Generally, if the blocks are very different, then this will occur.
Blocking eliminates the variabiliy associated with the nuisance variable that they represent. If the mean burning times of the two flames differ by as much as 2 minutes, find the power of the test. What sample size would be required to detect an actual difference in mean burning time of 1 minute with a power of at least 0. Rework this problem assuming that the two population variances are unknown but equal. There is no difference in the machines. The P-value for this anlysis is 0. The confidence interval is This interval contains 0.
There is no difference in machines. If the mean fill volume of the two machines differ by as much as 0. What sample size could result in a power of at least 0. Four different mixing techniques can be used economically. There is only a 0. Mixing technique has an effect. The results agree with the graphical method for this experiment.
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What conclusion would you draw about the validity of the normality assumption? There is nothing unusual about the normal probability plot of residuals. Comment on the plot. There is nothing unusual about this plot. Predicted Design-Expert automatically generates the scatter plot. The plot below also shows the sample average for each treatment and the 95 percent confidence interval on the treatment mean.
The mean of Treatment 4 is different than the means of Treatments 1, 2, and 3. However, the mean of Treatment 2 is not different from the means of Treatments 1 and 3 according to the Tukey method, they were found to be different using the graphical method and the Fisher LSD method. Find a 95 percent confidence interval on the mean tensile strength of the portland cement produced by each of the four mixing techniques. Also find a 95 Does this aid in interpreting the results of the experiment?
Strength is usually affected by the percentage of cotton used in the blend of materials for the fiber. Perhaps a little petulant, Fisher wondered whether Bristol had simply gotten lucky and guessed correctly all eight times. He worked out the math for this possibility and realized the odds were 1 in So she probably could taste the difference. But even then he couldn't stop thinking about the experiment. What if she'd gotten just one cup wrong out of eight? He reran the numbers and found the odds of her guessing "only" seven cups correctly dropped from 1 in 70 to around 1 in 4.
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In other words, accurately identifying seven of eight cups meant she could probably taste the difference, but he'd be much less confident in her ability-and he could quantify exactly how much less confident. Furthermore, that lack of confidence told Fisher something: the sample size was too small. So he began running more numbers and found that 12 cups of tea, with 6 poured each way, would have been a better trial. An individual cup would carry less weight, so one data point wouldn't skew things so much.
Other variations of the experiment occurred to him as well for example, using random numbers of tea-first and milk-first cups , and he explored these possibilities over the next few months. Now this might all sound like a waste of time. After all, Fisher's boss wasn't paying him to dink around in the tearoom. But the more Fisher thought about it, the more the tea test seemed pertinent. In the early s there was no standard way to conduct scientific experiments: controls were rare, and most scientists analyzed data crudely. Fisher had been hired to design better experiments, and he realized the tea test pointed the way.
However frivolous it seemed, its simplicity clarified his thinking and allowed him to isolate the key points of good experimental design and good statistical analysis. He could then apply what he'd learned in this simple case to messy real-world examples-say, isolating the effects of fertilizer on crop production.
Fisher published the fruit of his research in two seminal books, Statistical Methods for Research Workers and The Design of Experiments. The latter introduced several fundamental ideas, including the null hypothesis and statistical significance, that scientists worldwide still use today. And the first example Fisher used in his book-to set the tone for everything that followed-was Muriel Bristol's tea test.
His intellectual acumen, however, did not insulate Fisher from the prejudices of his time when it came to class, race, and colonialism. Fisher was a well-known eugenicist and was steadfast in those beliefs throughout his life. When, in the aftermath of World War II, UNESCO formed a coalition of scientists to wrestle with Nazi science and provide the scientific backbone for the universal condemnation of racism, Fisher was among those who officially objected to what he saw as the project's "well-intentioned" but misguided mission, affirming his belief that groups differed " in their innate capacity for intellectual and emotional development.
But such convictions have done little to tarnish Fisher's legacy. He became a legend in biology for helping to unite the gene theory of Gregor Mendel with the evolutionary theory of Charles Darwin. But his biggest contribution to science remains his work on experimental design. The reforms he introduced are so ubiquitous that they're all but invisible nowadays-the sign of a true revolution. By Sam Kean August 6, Detail of an advertisement for Dutch tea company Van Nelle, ca.
Wikimedia Commons. In offering his colleague a cup of tea, Ronald Fisher was just being polite. He had no intention of kicking up a dispute much less remaking modern science. Apply effective organizational and presentational skills. Demonstrate general but incomplete command of knowledge and skills required for attaining most of the course learning outcomes. Show evidence of some analytical and critical abilities and logical thinking, and ability to apply knowledge to most familiar situations.
Apply moderately effective organizational and presentational skills. Demonstrate partial but limited command of knowledge and skills required for attaining some of the course learning outcomes. Show evidence of some coherent and logical thinking, but with limited analytical and critical abilities. Show limited ability to apply knowledge to solve problems. Apply limited or barely effective organizational and presentational skills.
Demonstrate little or no evidence of command of knowledge and skills required for attaining the course learning outcomes.