1. Consider a manufacturing process that is producing hypodermic needles that will be used for blood donations. These needles need to have a diameter of 1.65 mm-too big and they would hurt the donor (even more than usual), too small and they would rupture the red blood cells, rendering the donated blood useless. Thus, the manufacturing process would have to be closely monitored to detect any significant departures from the desired diameter. During every shift, quality control personnel take a random sample of several needles and measure their diameters. If they discover a problem, they will stop the manufacturing process until it is corrected. For now, suppose that a "problem" is when the sample average diameter turns out to be statistically significantly different from the target of 1.65 mm.
a. Identify the variable of interest and whether the variable is categorical or quantitative.
b. Write the appropriate hypotheses using appropriate symbols to test whether the average diameter of needles from the manufacturing process is different from the desired value.
c. Suppose that the most recent random sample of 35 needles have an average diameter of 1.64 mm and a standard deviation of 0.07 mm. Assign appropriate symbols to these numbers.
d. Suppose that the diameters of needles produced by this manufacturing process have a bell-shaped distribution. Sketch the distribution of the average diameter of samples of 35 needles, assuming that the process is not malfunctioning.
Be sure to clearly label the axis of the graph and provide values for what you think the mean and standard deviation for this distribution should be.
2. According to a 2011 report by the United States Department of Labor, civilian Americans spend 2.75 hours per day watching television. A faculty researcher, Dr. Sameer, at California Polytechnic State University (Cal Poly) conducts a study to see whether a different average applies to Cal Poly students. Suppose that for a random sample of 100 Cal Poly students, the mean and standard deviation of hours per day spent watching TV turns out to be 3.01 and 1.97 hours, respectively. The data were used to fi nd a 95% confidence interval: (2.619, 3.401) hours/day. Which of the following are valid interpretations of the 95% confidence interval? For each of the following, statements, say whether it is VALID or INVALID.
a. About 95% of all Cal Poly students spend between 2.619 and 3.401 hours/day watching TV.
b. There is a 95% chance that, on average, Cal Poly students spend between 2.619 and 3.401 hours/day watching TV.
c. We are 95% confident that, on average, these 100 Cal Poly students spend between 2.619 and 3.401 hours/day watching TV.
d. In the long run, 95% of the sample means will be between 2.619 and 3.401 hours.
e. None of the above.