19 Consider a manufacturing process that is producing hypodermic needles that will be used for blood donations. These needles need to have a diameter...

2.2.19 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.

3.3.4 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.