The breaking strength of a fiber is required to be at least 150 psi. Past experience has indicated that the standard deviation of breaking strength is σ = 3 psi.
A random sample of four specimens is tested, and the results are y1 = 145, y2 = 153, y3 = 150, and y4, = 147.
(a) State the hypotheses that you think should be tested in this experiment.
(b) Test these hypotheses using a = 0.05. What are your conclusions?
(c) Find the P-value for the test in part (b).
(d) Construct a 95 percent confidence interval on the mean breaking strength.
Consider the shelf life data in Problem 2-5. Can shelf life be described or modeled adequately by a normal distribution? What effect would violation of this assumption have on the test procedure you used in solving Problem 2-5?
The shelf life of a carbonated beverage is of interest. Ten bottles are randomly selected and tested, and the following results are obtained:
Three brands of batteries are under study. It is suspected that the lives (in weeks) of the three brands are different. Five batteries of each brand are tested with the following results:
(a) Are the lives of these brands of batteries different?
(b) Analyze the residuals from this experiment.
(c) Construct a 95 percent confidence interval estimate on the mean life of battery brand 2. Construct a 99 percent confidence interval estimate on the mean difference between the lives of battery brands 2 and 3.
(d) Which brand would you select for use? If the manufacturer will replace without charge any battery that fails in less than 85 weeks, what percentage would the company expect to replace?
An article in Environment International (Vol. 18, No. 4, 1992) describes an experiment in which the amount of radon released in showers was investigated. Radon-enriched water was used in the experiment, and six different orifice diameters were tested in shower heads. The data from the experiment are shown in the following table:
(a) Does the size of the orifice affect the mean percentage of radon released? Use α = 0.05.
(b) Find the P-value for the F statistic in part (a).
(c) Analyze the residuals from this experiment.
(d) Find a 95 percent confidence interval on the mean percent of radon released when the orifice diameter is 1.40.
(e) Construct a graphical display to compare the treatment means as described in Section 3-5.3 What conclusions can you draw?
The response time in milliseconds was determined for three different types of circuits that could be used in an automatic valve shutoff mechanism. The results are shown in the following table:
(a) Test the hypothesis that the three circuit types have the same response time. Use α = 0.01.
(b) Use Tukey’s test to compare pairs of treatment means. Use α = 0.01.
(c) Use the graphical procedure in Section 3-5.3 to compare the treatment means. What conclusions can you draw? How do they compare with the conclusions from part (b)?
(d) Construct a set of orthogonal contrasts, assuming that at the outset of the experiment you suspected the response time of circuit type 2 to be different from the other two.
(e) If you were the design engineer and you wished to minimize the response time, which circuit type would you select?
(f) Analyze the residuals from this experiment. Are the basic analysis of variance assumptions satisfied?
An article in the ACI Materials Journal (Vol. 84,1987, pp. 213-216) describes several experiments investigating the rodding of concrete to remove entrapped air. A 3-inch × 6-inch cylinder was used, and the number of times this rod was used is the design variable. The resulting compressive strength of the concrete specimen is the response. The data are shown in the following table:
(a) Is there any difference in compressive strength due to the rodding level? Use α = 0.05.
(b) Find the P-value for the F statistic in part (a).
(c) Analyze the residuals from this experiment. What conclusions can you draw about the underlying model assumptions?
(d) Construct a graphical display to compare the treatment means as described in Section 3-5.3.
A manufacturer of television sets is interested in the effect on tube conductivity of four different types of coating for color picture tubes. The following conductivity data are obtained:
(a) Is there a difference in conductivity due to coating type? Use α = 0.05.
(b) Estimate the overall mean and the treatment effects.
(c) Compute a 95 percent confidence interval estimate of the mean of coating type 4. Compute a 99 percent confidence interval estimate of the mean difference between coating types 1 and 4.
(d) Test all pairs of means using the Fisher LSD method with α = 0.05.
(e) Use the graphical method discussed in Section 3-5.3 to compare the means. Which coating type produces the highest conductivity?
(f) Assuming that coating type 4 is currently in use, what are your recommendations to the manufacturer? We wish to minimize conductivity.
An experiment was run to determine whether four specific firing temperatures affect me density of a certain type of brick. The experiment led to the following data:
(a) Does the firing temperature affect the density of the bricks? Use α = 0.05.
(b) Is it appropriate to compare the means using Duncan’s multiple range test (for ex-ample) in this experiment?
(c) Analyze the residuals from this experiment. Are the analysis of variance assumptions satisfied?
(d) Construct a graphical display of the treatment as described in Section 3-5.3. Does this graph adequately summarize the results of the analysis of variance in part (a)?
The tensile strength of portland cement is being studied. Four different mixing techniques can be used economically. The following data have been collected:
(a) Test the hypothesis that mixing techniques affect the strength of the cement. Use α = 0.05.
(b) Construct a graphical display as described in Section 3-5.3 to compare the mean tensile strengths for the four mixing techniques. What are your conclusions?
(c) Use the Fisher LSD method with α = 0.05 to make comparisons between pairs of means.
(d) Construct a normal probability plot of the residuals. What conclusion would you draw about the validity of the normality assumption?
(e) Plot the residuals versus the predicted tensile strength. Comment on the plot.
(f) Prepare a scatter plot of the results to aid the interpretation of the results of this experiment.
In semiconductor manufacturing wet chemical etching is often used to remove silicon from the backs of wafers prior to metalization. The etch rate is an important characteristic of this process. Two different etching solutions are being evaluated. Eight randomly selected wafers have been etched in each solution and the observed etch rates (in mils/min) are shown below.
(a) Do the data indicate that the claim that both solutions have the same mean etch rate is valid? Use α = 0.05 and assume equal variances.
(b) Find a 95 percent confidence interval on the difference in mean etch rates.
(c) Use normal probability plots to investigate the adequacy of the assumptions of nor-mality and equal variances.