 # Problems 1. The measured data on the thickness of a metal layer fr...

## Question

Problems
1. The measured data on the thickness of a metal layer from a vapor deposition study in Å in an instrument is: 48530, 48980, 50210, 49860, 48650, 49560, 49270, 48850, 49320, 48680.
The instrument has a range of 100,000 with a full-scale accuracy of 1.5%.
a. Present the descriptive statistics for this data
b. Calculate the systematic uncertainty, random uncertainty, total uncertainty, random uncertainly of the mean, and total uncertainty of the mean from the instrument.

2. The inside diameters of bearings used in an aircraft landing gear assembly are known to have a standard deviation of σ = 0.002 cm. A random sample of 15 bearings has an average inside diameter of 8.2535 cm.
a. Test the hypothesis that the mean inside bearing diameter is 8.25 cm. Use a two-sided alternative and α = 0.05.
b. Find the P-value for this test.
c. Construct a 95% two-sided confidence interval on the mean bearing diameter
d. Repeat (a,b,c) above for α = 0.1. Compare with results for α = 0.05.
e. Repeat (a,b,c) above for α = 0.01. Compare with results for α = 0.05.

3. A machine is used to fill containers with a liquid product. Fill volume can be assumed to be normally distributed. A random sample of ten containers is selected, and the net contents (oz) are as follows: 12.03, 12.01, 12.04, 12.02, 12.05, 11.98, 11.96, 12.02, 12.05, 11.99.
a. Suppose that the manufacturer wants to be sure that the mean net contents exceeds 12 oz. What conclusions can be drawn from the data. Use a = 0.01.
b. Construct a 95% two-sided confidence interval on the mean fill volume.
c. Does the assumption of normality seem appropriate for the fill volume data?
d. Repeat (a,b,c) above for a = 0.05. Compare with results for a = 0.01.

4. Two machines are used for filling glass bottles with a soft-drink beverage. The filling processes have known standard deviations s1 = 0.010 L, and s2 = 0.015 L, respectively. A random sample of n1 = 25 bottles from machine 1 and n2 = 20 bottles from machine 2, results in average net contents of x1= 2.04 L, and x2 = 2.07 L.
a. Test the hypothesis that both machines fill to the same net contents, using a = 0.05. What are your conclusions?
b. Find the P-value for this test.
c. Construct a 95% two-sided confidence interval on the mean fill volume.

5. Two quality control technicians measured the surface finish of a metal part, and the data is tabulated. Assume that the measurements are
normally distributed.
a. Test the hypothesis that the mean surface finish measurements made by the two technicians are equal. Use a = 0.05, and assume equal variances
b. Assuming that the variances are equal, construct a 95% confidence interval on the mean difference in surface-finish measurements.
Problem 5
T1      T2
1.45   1.54
1.37   1.41
1.21   1.56
1.54   1.37
1.48   1.20
1.29   1.31
1.34   1.27
1.35

6. Two different hardening processes – (1) saltwater quenching and (2) oil quenching – are used on samples of a particular type of metal alloy. The results are tabulated. Assume that hardness is normally distributed.
a. Test the hypothesis that the mean hardness for the saltwater quenching process equals the mean hardness for the oil quenching process. Use a = 0.05, and assume equal variances.
b. Assuming that the variances are equal, construct a 95% confidence interval on the mean difference in mean hardness
c. Does the assumption of normality seem appropriate for this data?

Problem 6
Saltwater Oil
145          152
150          150
153          147
148          155
141          140
156          146
146          158
154          152
139          151
148          143

7. The results of an experiment to investigate the low-pressure vapor deposition of polysilicon. The reactor has several wafer positions,
and four of these positions are selected at random. The response variable is film thickness uniformity. Three replicates of the experiment were run, and data is presented.
a. Is there a difference in the wafer positions? Use the analysis of variance, and a = 0.05?
b. Estimate the variability due to wafer positions.
c. Estimate the random error component
d. Analyze the results from this experiment and comment on model adequacy.
Problem 7
Position          Uniformity
1                2.76   5.67   4.49
2                1.43   1.70   2.19
3                2.34   1.97   1.47
4                0.94   1.36   1.65

8. An experiment to determine the effect of C2F6 flow rate on etch uniformity on Si wafer is presented in a table. Three flow rates are tested, and the resulting uniformity (in %) is observed for six test units at each flow rate.
a. Does C2F6 flow rate affect etch uniformity? Use the analysis of variance, and a = 0.05?
b. Construct a box plot of the etch uniformity data. Use this plot, together with ANOVA results, to determine which gas flow rate would be best in terms of etch uniformity (a small % is best).
c. Plot the residuals vs. predicted C2F6 flow. Interpret this plot.
d. Does the normality assumption seem reasonable in the problem?
Problem 8
Observations
Flow    1    2    3    4    5    6
125    2.7   2.6 4.6 3.2 3.0 3.8
160    4.6   4.9 5.0 4.2 3.6 4.2
200    4.6   2.9 3.4 3.5 4.1   5.1

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