Applied Statistics for Engineers and Scientists - Practice Problems 7 | STAT 541, Assignments of Statistics

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Practice Problems # 07

1. Five measurements are taken of the octane rating for a particular type of gasoline. Assume these are a simple random sample from the population of this type of gasoline, and that the distribution of measurements is a normally distributed population. The results (in%) are 87.0, 86.0, 86.5, 88.0, 85.3: find a 99% two-sided tolerance interval on 90% of the population.
2. For the data in problem number 3, find a 98% confidence interval for the standard deviation of the population of measurements.
3. Compounds of mercury and mercury ions are discharged into the atmosphere when coal is burned in large quantities, as in a power generation station. These are then transported to the land surface and water ways by rain, and of course, eventually make their way into the water supply. Let μ denote the true average mercury level in the James River as measured in ppm. A value of 10 ppm is considered the dividing line between safe and unsafe water levels of mercury. Would you recommend testing H 0 : μ = 10 versus Ha: μ > 10 or H 0 : μ = 10 versus Ha: μ < 10? Explain your reasoning. Think about the consequences of the type I and II errors for each possibility.
4. A company manufacturers piano wire. A measure of the quality of the wire can be determined by the amount of extension of the individual wires under a load of 30 N. Let μ represent the mean extension for the population of piano wires and it is known from past history that σ = 0.020 mm. On a particularly warm day, a simple random sample of 65 lengths of wire produced a sample mean of 1.102 mm.

a. Test the hypothesis that the mean length is equal to 1.1 versus the alternative that the mean length is greater than 1.1 at the 5% significance level. b. Find the p-value for this test. c. Either the mean extension, μ, for this days production is greater than 1.1 mm, or the sample is in the most extreme ______________ % of its distribution. d. For a level 0.05 test, what is β(1.15), the probability of a type II error when μ = 1.15?

5. A particular type of gasoline is supposed to have a mean octane rating greater than 90%. Five measurements are taken of the octane rating, as follows: 90.1 88.8 89.5 91.0 92. Can you conclude that the mean octane rating is greater than 90%? Assume the octane measurement variable is normally distributed and use a significance level of 0.01.
6. If p-value = 0.013, which is the best conclusion? Select only one. a. H 0 is definitely false. b. H 0 is definitely true. c. There is a 1.3% probability that H 0 is true. d. H 0 might be true, but it’s unlikely. e. H 0 might be false, but it’s unlikely. f. H 0 is plausible.