See Solution: The Quality control department of a paper company measures - #80118

Problem 1

The Quality control department of a paper company measures the brightness (a measure of the reflectance) of finished paper on a periodic basis throughout the day. Two instruments that are available to measure the paper specimens are subject to error, but they can be adjusted so that the mean readings for a control paper specimen are the same for both instruments. Suppose you are concerned about the precision of the two instruments – namely that instrument 2 is less precise than instrument 1. To check this theory, five measurements of a single paper sample are made on both instruments. The data are shown below. Do the data provide sufficient evidence to indicate that instrument 2 is less precise than instrument 1? Assuming normality, you must be at least 90 confident in your conclusion before taking any action, if necessary.

Instrument 1	Instrument 2
105	100
111	110
102	105
102	103
106	105

Parameter(s) to be Tested:
H₀:
H_a:
Critical Value:
Rejection Region:
Value of the Test Statistic:
Conclusion (a statement):

Problem 2

Pascal is a high level programming language used frequently in microprocessors. An experiment was conducted to investigate the proportion of Pascal variables that are array variables, which are less efficient in terms of execution time. Twenty variables are randomly selected from a set of Pascal programs and Y, the number of array variables, is recorded. Suppose we want to test the hypothesis that Pascal is more efficient language than Algol, in which 20% of the variables are array variables. That is, we will test H₀: p =.20 against H_a: p > .20 , where p is the probability of observing an array variable on each trial (Assume that the 20 trials are independent).

a) Find α for the rejection (Y ≥ 8)

b) Find α for the rejection (Y ≥ 5)

c) Find β for the rejection Y ≥ 8 if p = 0.5 (Note: Past experience has shown that approximately half the variables in most Pascal programs are array variables.)

d) Find β for the rejection Y ≥ 5 if p = .5

e) Which of the rejection regions Y ≥ 8 or Y ≥ 5 is more desirable if you want to minimize the probability of a Type I error? Type II error?

f) Find the rejection region of the form Y ≥ α so that α is approximately equal to .01.

a) α (Y ≥ 8) =
b) α (Y ≥ 5) =
c) β‌ (Y ≥ 8│p = 0.5) =
d) β‌ (Y ≥ 5│p = 0.5)
e) If want to minimize probability of Type I error (why?):
e) If want to minimize probability of Type II error (why?):
f) For α = 0.01, Y =

Problem 3

Cooling pipes at three nuclear power plants are investigated for deposits that would inhibit the flow of water. From 30 randomly selected spots at each plant, 13 from the first plant, 8 from the second plant, and 19 from the third were clogged.

a) At a 0.05 level, does the data indicate that the proportion of clogged cooling pipes is dependent of the plant investigated?

b) Construct the confidence intervals for the three probabilities of being clogged.

Parameter(s) to be Tested:
H₀:
H_a:
Critical Value:
Rejection Region:
Value of the Test Statistic:
Conclusion (a statement):

Probability of being clogged	Confidence Interval (a, b)
At Plant I
At Plant II
At Plant III

Problem 4

Suppose the fraction of defective modems shipped by a data-communications vendor has an approximate beta distribution with α = 5 and β = 21.

a) Find the mean and variance of the fraction of defective modems per shipment

b) What is the probability that a randomly selected shipment will contain at least 30% defective?

c) What is the probability that a randomly selected shipment will contain no more than 5% defective?

Problem 5

Past data indicate that the variance of measurements made on sheet metal stampings by experienced quality-control inspectors is 0.25 square inch. Such measurements made by an inexperienced inspector could have too large a variance (perhaps because of inability to read instruments properly) or too small a variance (perhaps because unusually high or low measurements are discarded). If a new inspector measures 101 stampings with a variance of 0.13 square inch, test at the 0.05 level of significance whether the inspector is making satisfactory measurements. Assume normality.

Parameter(s) to be Tested:
H₀:
H_a:
Critical Value:
Rejection Region:
Value of the Test Statistic:
Conclusion (a statement):

Problem 6

The FDA conducted a study to evaluate the effectiveness of a drug it recently developed to reduce the level of Total Blood Cholesterol. In the study, 20 persons with a high level cholesterol (the desirable level is less than 200 milligrams per deciliter of blood (mg/dL)) were administered the newly developed pill. The pharmaceutical company that manufactures the pill claims that patients with high cholesterol level will experience a reduction of at least 50 mg/dL using this pill. The results are shown in the table below. Does the data support the company’s claim? Test at α = 0.05. Assume normality.

Parameter(s) to be Tested:
H₀:
H_a:
Critical Value:
Rejection Region:
Value of the Test Statistic:
Conclusion (a statement):
p-value
p-value interpretation (statement):