## Instructions

This homework is due on Tuesday, December 12th at 2pm (the start of class). Please turn in all your work. This homework covers linear regression, matched pairs and tests on several related samples.

• Calculations by hand: Throughout this homework, and beyond, “by hand” means either (1) you utilize quantile/distribution tables, and/or Gaussian approximations, as appropriate, and otherwise do all of your calculations with pen and paper (and a calculator); or (2) you write code, say in R, building up all of the steps yourself, i.e., not using a library function that automates the entire procedure (see next bullet).

• Using a software library: Through this the homework, and beyond, “using a software library” means you can feed your data into a built-in function, like t.test and binom.test in R, and interpret the output as appropriate. Be sure to provide details on the library you used, how you used it, what the output was, and what it means.

### Problem 1: Meter bias (15 points)

Suppose that an engineer is interested in testing the bias in a pH meter. Data are collected on a neutral substance (pH=7.0). A sample of the measurements were taken with the data as follows:

pH <- c(7.07, 7.00, 7.10, 6.97, 7.00, 7.03, 7.01, 7.01, 6.98, 7.08)

Is the meter biased?

1. (10 pts) Conduct the test “by hand”.
2. (5 pts) Perform the test “using a software library”.

### Problem 2: Agent Orange (15 points)

A study reported the levels of the dioxin TCDD of 20 Massachusetts Vietnam veterans who were possibly exposed to Agent Orange. The amount of TCDD in plasma and fat tissue are listed in the table on the next page. Is there enough evidence to conclude that the levels in the fat tissue are higher than in the plasma.

veteran <- data.frame(
plasma=c(2.5, 3.1, 2.1, 3.5, 7.2, 1.8, 6.0, 3.0, 36.0, 4.7, 6.9,
3.3, 4.6, 1.6, 7.3, 1.8, 20.0, 2.0, 2.5, 2.3),
fattissue=c(4.9, 5.9, 4.4, 6.9, 4.0, 4.2, 10.0, 5.5, 41.0, 4.4,
7.0, 2.9, 4.6, 1.4, 7.7, 1.1, 11.0, 2.5, 2.3, 2.5))
1. (10 pts) Conduct the test “by hand”.
2. (5 pts) Perform the test “using a software library”.

### Problem 3: DUI (20 points)

A random sample consisting of 10 people who drove automobiles was selected to see if alcohol affected reaction time. Each driver’s reaction time was measured in a laboratory before and after drinking a specified amount of a beverage containing alcohol. The reaction times in seconds were as follows:

dui <- data.frame(before=c(0.65, 0.59, 0.78, 0.67, 0.65, 0.76, 0.61, 0.86, 0.74, 0.88),
after=c(0.72, 0.60, 0.78, 0.66, 0.68, 0.77, 0.72, 0.86, 0.72, 0.97))
1. (5 pts) Construct a plot or plots summarizing these data in a meaningful way.
2. (5 pts) Calculate informative numerical summaries.
3. (5 pts) Suppose you are interested in determining whether alcohol affects reaction time. Without conducting the test, what do you suspect?
4. (5 pts) Perform the test. You need to provide hypotheses, test statistic value, $$p$$-value and conclusions.

### Problem 4: Machine assemply (25 points)

Four different machines, $$M_1$$, $$M_2$$, $$M_3$$ and $$M_4$$, are being considered for the assembly of a particular product. It is decided that 6 different operators are to be used in order to compare the performance of the machines. Do you believe all machines perform the same? The amount of time (in seconds) needed to assemble the product is recorded in the following table:

Machine \ Operator 1 2 3 4 5 6
1 42.6 39.3 39.6 39.9 42.9 43.6
2 39.8 40.1 40.5 42.3 42.5 43.1
3 40.2 40.5 41.3 43.4 44.9 45.1
4 41.3 42.2 43.5 44.2 45.9 42.3
1. (5 pts) Which test would you use and why?
2. (15 pts) Perform the test “by hand”, including multiple comparisons if necessary.
3. (5 pts) Perform the test in a. “using a software library”.

### Problem 5: Course difficulty (25 points)

The following data represent the final grades obtained by 5 students in mathematics, French and biology. Are all courses equal in difficulty?

Student \ Subject Math French Biology
1 68 73 61
2 83 91 86
3 72 77 66
4 55 77 66
5 92 75 87
1. (5 pts) Which test would you use and why?
2. (15 pts) Perform the test “by hand”, including multiple comparisons if necessary.
3. (5 pts) Perform the test in a. “using a software library”.