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Assignment # 02
Hypothesis Testing for mean
Syed Tauqeer Ahmed Hashmi
1.
When a manufacturing process is operating properly, the mean length of a certain part is known to be 6.175 inches, and lengths are normally distributed. The standard deviation of this length is 0.0080 inches. If a sample consisting of 6 items taken from current production has a mean length of 6.168 inches, is there evidence at the 5% level of significance that some adjustment of the process is required? 2.
A taxi company has been using Brand A tires, and the distribution of kilometers to wear-out has been found to be approximately normal with mean =114,000 and s.d =11,600. Now it tries 12 tires of Brand B and finds a sample mean 117,200. Test at the 5% level of significance to see whether there is a significant difference (positive or negative) in kilometers to wear-out between Brand A and Brand B. Assume the standard deviation is unchanged. 3.
The average daily amount of scrap from a particular manufacturing process is 25.5 kg with a standard deviation of 1.6 kg. A modification of the process is tried in an attempt to reduce this amount. During a 10-day trial period, the kilograms of scrap produced each day were: 25.0, 21.9, 23.5, 25.2, 22.0, 23.0, 24.5, 25.0, 26.1, and 22.8. From the nature of the modification, no change in day-to-day variability of the amount of scrap will result. The normal distribution will apply. A first glance at the figures suggests that the modification is effective in reducing the scrap level. Does a significance test confirm this at the 1% level? 4.
The standard deviation of a particular dimension on a machine part is known to be 0.0053 inches. Four parts coming off the production line are measured, giving readings of 2.747, 2.740, 2.750 and 2.749 inches. The population mean is supposed to be 2.740 inches. The normal distribution applies. Is the sample mean significantly larger than 2.740 inches at the 1% level of significance? 5.
The outlet stream of a continuous chemical reactor is sampled every thirty minutes and titrated. Extensive records of normal operation show the concentration of component A in this stream is approximately normally distributed with mean 41.2 g/L and standard deviation 0.90 g/L. a)
What is the probability that the concentration of component A in this stream will be more than 42.3 g/L? b)
Five determinations of concentration of component A are made. If the mean of these five concentrations is more than 42.3 g/L, action is taken. State the null hypothesis and the alternative hypothesis that fit the test? 6.
A manufacturer produces a special alloy steel with an average tensile strength of 25,800 psi. The standard deviation of the tensile strength is 300 psi. Strengths are approximately normally distributed. A change in the composition of the alloy is tried in an attempt to increase its strength. A sample consisting of eight specimens of the new composition is tested. Unless an
increase
in the strength is significant at the 1% level, the manufacturer will return to the old composition. Standard deviation is not affected. a)
If the mean strength of the sample of eight items is 26,100 psi, should the manufacturer continue with the new composition? b)
What is the minimum mean strength that will justify continuing with the new composition?
Assignment # 02
Hypothesis Testing for mean
Syed Tauqeer Ahmed Hashmi
7.
Noise levels in the cabs of a large number of new farm tractors were measured ten years ago and were found to vary about a mean value of 76.5 decibels (db) with a variance of 72.43 db2. A
researcher conducted a survey of this year’s new
tractors to determine whether or not tractor cab manufacturers have been successful in developing quieter cabs. In her final report, the researcher stated that the mean noise level in the cabs she studied was 74.5 db, and she concluded that there was only 12% probability of getting results at least this far different if there was no real reduction in noise level. Calculate the number of cabs that the researcher must have surveyed in order to have drawn this conclusion. 8.
Jack Spratt is in charge of quality control of the concrete poured during the construction of a certain building. He has specimens of concrete tested to determine whether the concrete strength is within the specifications; these call for a mean concrete strength of no less than 30 MPa. It is known that the strength of such specimens of concrete will have a standard deviation of 3.8 MPa and that the normal distribution will apply. Mr. Spratt is authorized to order the removal of concrete which does not meet specifications. Since the general contractor is a burly sort, Mr. Spratt would like to avoid removing the concrete when the action is not justified. Therefore, the probability of rejecting the concrete when it actually meets the specification should be no more than 1%. What size sample should Mr. Spratt use if a sample mean 10% less than the specified mean strength will cause rejection of the concrete pour? State the null hypothesis and alternative hypothesis. 9.
The following data were obtained for the calibration of the Ruska dead weight gauge used with our Burnett PVT Apparatus. The weights corresponding to the 1.000 PSI loading had the following apparent masses: 26.03570; 26.03581; 26.03529; 26.03573; 26.03575; 26.03551; 26.03588; 26.03586; 26.03599; 26.03533; 26.03570. Can we say that the average apparent mass does not exceed 26.5? 10.
Fifty determinations of a certain concentration yielded the following values: 54.20; 51.73; 52.56; 53.55; 56.15; 57.50; 54.25; 54.46; 53.08; 53.82; 54.15; 53.10; 51.56; 53.43; 53.77; 55.88; 54.96; 58.51; 54.65; 55.13; 51.12; 53.73; 55.01; 55.57; 53.95; 53.39; 54.30; 52.89; 57.35; 55.77; 52.22; 54.55; 56.78; 56.00; 57.27; 54.89; 57.05; 56.25; 56.35; 56.52; 56.91; 52.35; 52.02; 52.94; 58.16; 57.73; 55.33; 54.13; 56.60; 55.21. Test the hypothesis H0 : µ = 55.0 with the 99% confidence level. 11.
Two chemical processes for manufacturing the same product are being compared under the same conditions. Yield from Process A gives an average value of 96.2 from six runs, and the estimated standard deviation of yield is 2.75. Yield from Process B gives an average value of 93.3 from seven runs, and the estimated standard deviation is 3.35. Yields follow a normal distribution. Is the difference between the mean yields statistically significant? Use the 5% level of significance, and show rejection regions for the difference of mean yields on a sketch
Assignment # 02
Hypothesis Testing for mean
Syed Tauqeer Ahmed Hashmi
12.
High sulfur content in steel is very undesirable, giving corrosion problems among other disadvantages. If the sulfur content becomes too high, steps have to be taken. Five successive independent specimens in a steel-making process give values of percentage sulfur of 0.0307, 0.0324, 0.0314, 0.0311 and 0.0307. Do these data give evidence at the 5% level of significance that the true mean percentage sulfur is above 0.0300? What is the 90% two-sided confidence interval for the mean percentage sulfur in the steel? 13.
Two companies produce resistors with a nominal resistance of 4000 ohms. Resistors from company A give a sample of size 9 with sample mean 4025 ohms and estimated standard deviation 42.6 ohms. A shipment from company B gives a sample of size l3 with sample mean 3980 ohms and estimated standard deviation 30.6 ohms. Resistances are approximately normally distributed. a.
At 5% level of significance, is there a difference in the mean values of the resistors produced by the two companies? b.
Is either shipment significantly different from the nominal resistance of 4000 ohms, use 0.05 level of significance. 14.
Two different types of evaporation pans are used for measuring evaporation at a weather station. The evaporation for each pan for 6 different days is as follows: At the 5% level of significance, is there a significant difference in the evaporation recorded by the two pans? Interaction between type of pan and weather variation from day to day can be neglected. 15.
Daily evaporation rates were measured on 20 successive days. Which of two types of evaporation pan would be used on a particular day was decided by tossing a coin. The mean daily evaporation for the 10 days on which Pan A was used was 19.10 mm, and the mean evaporation on the 10 days on which Pan B was used was 17.24 mm. The variance estimated from the sample from Pan A is 7.72 mm
2
, and the variance estimated from the sample from Pan B is 5.36 mm
2
. Assuming that these two estimates of variance are equal, does the experimental evaporation pan, Pan A, give significantly higher evaporation rates than the standard pan, Pan B, at the 1% level of significance?
Assignment # 02
Hypothesis Testing for mean
Syed Tauqeer Ahmed Hashmi
Bonus Questions:
1.
The amount of fluoride in the local water supply was determined by the four colorimetric methods in a comparative study A, B, C and D. Five replications were made for each test. To preclude bias from variations in the sample over the time required for the analysis, all samples were taken from a single 10-gal carboxy of water. The results in ppm are: A: 2; 3; 6; 5; 4; B: 5; 4; 4; 2; 3; C: 1; 3; 2; 4; 4; D: 2; 1; 1; 2; 1. a.
Are the methods equivalent? Use the 5% significance level. 2.
The conductivity of four different coatings on cathode tubes was tested. As only four types of coatings were tested, we had a one-way experiment on four levels. The mentioned levels were qualitative as we had no quantitative measure for coating types. Five cathode tubes were tested for each coating. The sequence of conductivity measurements was completely random. The obtained results are given in the following table: Coating I II III IV 56 64 45 42 55 61 46 39 62 50 45 45 59 55 39 43 60 56 43 41 If we subtract 50 from each value we shall obtain coded values, which to a great extent will simplify the arithmetic and has no influence on the F-statistic.
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