Department of Statistics, Virginia Tech

Below we review via several examples, but first I want to convey what the exam will look like (predominantly; but variations can be expected). There will be a style of question that will be repeated several times. And you will have a table to consult with relevant calculations. I won’t tell you which will be relevant for what, and there will be a few red herrings, but hopefully after today you’ll get the idea.

On the exam you may have:

- One sheet of notes, front and back, that you produced yourself.
- Calculator. (This is not really optional.)
- Writing device. I will provide all paper.

Some notes on performing calculations.

- For all two-sided tests, please calculate the \(p\)-value as \[ \varphi = 2 \times \min\{ \mathbb{P}(T \leq t), \mathbb{P}(T \geq t)\}. \]
- All hypothesis tests default to an \(\alpha = 0.05\) level unless otherwise stated; CIs may specify a different \(\alpha\).
- For tolerance limits, please use the approximations given in class (with reference to \(x_{1-\alpha}\) quantiles. The true values are not computable without R.

Several questions on the exam will involve tests of hypotheses and will be comprised of the following parts.

To answer part c. in particular you will need to consult calculations on the cumulative distribution function (CDF) for various random variables. You will have a table on your exam that looks something like this. They are arranged in an alphanumeric order for ease of reference. Most will be unnecessary for your calculations. At the bottom there are calculations on the inverse CDF, i.e., the quantile function, which will be useful for questions on intervals.

RV/distribution | probability statement | value |
---|---|---|

\(Z \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(Z \leq -1.0845)\) | 0.1390716 |

\(Z \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(Z \leq -1.390575)\) | 0.0821772 |

\(Z \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(Z \leq 1.390575)\) | 0.9178228 |

\(Z \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(Z \leq 0.9886)\) | 0.8410057 |

\(Z \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(Z \leq 1.0845)\) | 0.8609284 |

\(Z \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(Z \leq 1.9568)\) | 0.9748145 |

\(T \sim \chi^2_2\) | \(\mathbb{P}(T \leq 2.8)\) | 0.753403 |

\(T \sim \chi^2_2\) | \(\mathbb{P}(T \leq 3.55)\) | 0.8305166 |

\(T \sim \chi^2_3\) | \(\mathbb{P}(T \leq 2.8)\) | 0.5765001 |

\(T \sim \chi^2_3\) | \(\mathbb{P}(T \leq 3.55)\) | 0.685665 |

\(T \sim \chi^2_3\) | \(\mathbb{P}(T \leq 14.98)\) | 0.9981662 |

\(T \sim \chi^2_3\) | \(\mathbb{P}(T \leq 17.6)\) | 0.9994682 |

\(T \sim \chi^2_3\) | \(\mathbb{P}(T \leq 7.7782)\) | 0.9491749 |

\(T \sim \chi^2_3\) | \(\mathbb{P}(T \leq 8.2152)\) | 0.9582326 |

\(T \sim \chi^2_4\) | \(\mathbb{P}(T \leq 8.2152)\) | 0.9159943 |

\(T \sim \chi^2_4\) | \(\mathbb{P}(T \leq 14.98)\) | 0.9952571 |

\(T \sim \chi^2_4\) | \(\mathbb{P}(T \leq 17.6)\) | 0.9985228 |

\(T \sim \chi^2_4\) | \(\mathbb{P}(T \leq 7.7782)\) | 0.8999507 |

\(T \sim \chi^2_5\) | \(\mathbb{P}(T \leq 4.778)\) | 0.5564302 |

\(T \sim \chi^2_5\) | \(\mathbb{P}(T \leq 8.58)\) | 0.8729645 |

\(T \sim \chi^2_6\) | \(\mathbb{P}(T \leq 4.778)\) | 0.4274147 |

\(T' \sim \mathrm{Wilcox}(3,4)\) | \(\mathbb{P}(T' \leq 0)\) | 0 |

\(T' \sim \mathrm{Wilcox}(3,4)\) | \(\mathbb{P}(T' \leq 1)\) | 0.0571429 |

\(T' \sim \mathrm{Wilcox}(3,4)\) | \(\mathbb{P}(T' \leq 2)\) | 0.1142857 |

\(T' \sim \mathrm{Wilcox}(3,4)\) | \(\mathbb{P}(T' \leq 3)\) | 0.2 |

\(T' \sim \mathrm{Wilcox}(4,5)\) | \(\mathbb{P}(T' \leq 0)\) | 0 |

\(T' \sim \mathrm{Wilcox}(4,5)\) | \(\mathbb{P}(T' \leq 1)\) | 0.015873 |

\(T' \sim \mathrm{Wilcox}(4,5)\) | \(\mathbb{P}(T' \leq 2)\) | 0.031746 |

\(T' \sim \mathrm{Wilcox}(4,5)\) | \(\mathbb{P}(T' \leq 3)\) | 0.0555556 |

\(T \sim \mathrm{MannWhitney}(3,4)\) | \(\mathbb{P}(T \leq 9)\) | 0.2 |

\(T \sim \mathrm{MannWhitney}(3,4)\) | \(\mathbb{P}(T \leq 10)\) | 0.3142857 |

\(T \sim \mathrm{MannWhitney}(3,4)\) | \(\mathbb{P}(T \leq 11)\) | 0.4285714 |

\(T \sim \mathrm{MannWhitney}(3,4)\) | \(\mathbb{P}(T \leq 12)\) | 0.5714286 |

\(T \sim \mathrm{MannWhitney}(4,5)\) | \(\mathbb{P}(T \leq 9)\) | 0 |

\(T \sim \mathrm{MannWhitney}(4,5)\) | \(\mathbb{P}(T \leq 10)\) | 0.0079365 |

\(T \sim \mathrm{MannWhitney}(4,5)\) | \(\mathbb{P}(T \leq 11)\) | 0.015873 |

\(T \sim \mathrm{MannWhitney}(4,5)\) | \(\mathbb{P}(T \leq 12)\) | 0.031746 |

\(T \sim \mathrm{SqRank}(8,9)\) | \(\mathbb{P}(T \leq 1106.5)\) | 0.9135335 |

\(T \sim \mathrm{SqRank}(8,9)\) | \(\mathbb{P}(T \leq 1107.5)\) | 0.9143974 |

\(T \sim \mathrm{SqRank}(9,10)\) | \(\mathbb{P}(T \leq 1036.75)\) | 0.303687 |

\(T \sim \mathrm{SqRank}(9,10)\) | \(\mathbb{P}(T \leq 1107.5)\) | 0.4055944 |

The questions below do not overlap with ones from lecture, although some of them do come from the book. We will try to answer them in the format of the template, above, and consult the table of calculations. So solutions will not be provided in R – we’ll do them on the chalkboard to best duplicate the exam setting.

Four different methods of growing corn were randomly assigned to a large number of different plots of land and the yield per acre was computed for each plot. The grand median over all plots was 89, and the rest of the 34 observations over the four plots can be summarized below. Is there evidence of a difference in yields as a result of the method used?

Method 1 | Method 2 | Method 3 | Method 4 | |
---|---|---|---|---|

\(>\) 89 |
6 | 3 | 7 | 0 |

\(\leq\) 89 |
3 | 7 | 0 | 8 |

A die was cast 600 times with the following results

Occurrence |
1 | 2 | 3 | 4 | 5 | 6 |

Frequency |
87 | 96 | 108 | 89 | 122 | 98 |

Is the die balanced?

Each of three basketball enthusiasts had devised his own system for predicting the outcomes of collegiate basketball games. Twelve games were selected at random, and each sportsman presented a prediction of the outcome of each game. After the games were played, the results were tabulated, using 1 for successful prediction and 0 for unsuccessful prediction. Based on the data tabulated below, are the three sportsman equally good at predicting the outcome of games?

Game | Sportsman 1 | Sportsman 2 | Sportsman 3 |
---|---|---|---|

1 | 1 | 1 | 1 |

2 | 1 | 1 | 1 |

3 | 0 | 1 | 0 |

4 | 1 | 1 | 0 |

5 | 0 | 0 | 0 |

6 | 1 | 1 | 1 |

7 | 1 | 1 | 1 |

8 | 1 | 1 | 0 |

9 | 0 | 0 | 1 |

10 | 0 | 1 | 0 |

11 | 1 | 1 | 1 |

12 | 1 | 1 | 1 |

A simple experiment was designed to see if flint from area A tended to have the same degree of hardness as flint in area B. Four sample pieces of flint were collected in area A, and five in area B. To determine which of two pieces of flint is harder they were rubbed against each other. The piece sustaining less damage was judged to be the harder of the two. The ranking of hardnesses may be summarized as follows.

Origin | Rank |
---|---|

A | 1 |

A | 2 |

A | 3 |

B | 4 |

A | 5 |

B | 6 |

B | 7 |

B | 8 |

B | 9 |

Are the two flints of equal hardness?

A blood bank kept a record of the rate of heartbeats for several blood donors. Using the data blow, is there evidence of differences in variation among the heartbeats of men and women. For convenience, ranks of the combined sample of \(U\) and \(V\) values defined by absolute differences between the samples and their within-sample averages is provided.

Men | Women | R(U) | R(V) |
---|---|---|---|

58 | 66 | 17 | 13 |

76 | 74 | 4 | 7 |

82 | 69 | 14 | 6 |

74 | 76 | 1.5 | 12 |

79 | 72 | 11 | 3 |

65 | 73 | 15 | 5 |

74 | 75 | 1.5 | 9 |

86 | 67 | 16 | 10 |

68 | 8 |

Twelve MBA graduates are studied to measure the strength of the relationship between their score on the GMAT, which they took prior to entering graduate school, and their grade point average while they were in the MBA program. Their GMAT scores and their GPAs are given below, along with the ranks and some auxiliary computations. Is there a tendency for high GPAs to be associated with high GMAT scores?

Student | GMAT (\(X\)) | GPA (\(Y\)) | \(R(X)\) | \(R(Y)\) | \([R(X) - R(Y)]^2\) |
---|---|---|---|---|---|

1 | 710 | 4.0 | 12 | 11.5 | 0.25 |

2 | 610 | 4.0 | 9.5 | 11.5 | 4 |

2 | 640 | 3.9 | 11 | 10 | 1 |

4 | 580 | 3.8 | 8 | 9 | 1 |

5 | 545 | 3.7 | 3 | 8 | 25 |

6 | 560 | 3.6 | 5 | 7 | 4 |

7 | 610 | 3.5 | 9.5 | 5 | 20.25 |

8 | 530 | 3.5 | 1 | 5 | 16 |

9 | 560 | 3.5 | 5 | 5 | 0 |

10 | 540 | 3.3 | 2 | 3 | 1 |

11 | 570 | 3.2 | 7 | 1.5 | 30.25 |

12 | 560 | 3.2 | 5 | 1.5 | 12.25 |

Four different contractors manufacture one type of chemical detection kit. All kits produced by all of the contractors are supposed to respond to the same toxic gases. A test is performed to see if this is the case. Ten kits are randomly selected from lots manufactured by each of the four contractors. The forty kits are put in a gas chamber under laboratory conditions for a specified length of time, and are then compared. The responses of the kits are different shades of color in a patch. The colors are ranked from pink to dark purple as follows

A | B | C | D | |
---|---|---|---|---|

19 | 18 | 25 | 7 | |

10 | 5 | 3 | 20.5 | |

4 | 28 | 32 | 23 | |

38 | 1 | 29 | 13 | |

33 | 15 | 6 | 16 | |

36 | 12 | 2 | 9 | |

39 | 27 | 30 | 8 | |

40 | 31 | 35 | 14 | |

37 | 20.5 | 34 | 17 | |

26 | 22 | 24 | 11 | |

sum |
282 | 179.5 | 220 | 138.5 |

Is there a difference in kits manufactured by the various contractors? The sum of the square of the ranks is 22139.5.

The questions below don’t follow the above format because they involve the calculation of a confidence interval or tolerance limit, so they are not tests of hypotheses. Entries from the bottom portion of the table above will be helpful.

One hundred married couples were interviewed, and the husband and wife were asked separately for their choice for the next US president, with the following results.

Husband \ Wife | Clinton | Trump | Other |
---|---|---|---|

Clinton |
12 | 22 | 6 |

Trump |
25 | 21 | 4 |

Other |
3 | 7 | 0 |

- Compute Cramér’s coefficient \(C_{\mathrm{cc}}\), and Pearson’s contingency coefficients(s) \(R_2\) and \(R_3\). In each case, specify the ranges that those statistics may take on.

- Eliminate the “Other” category and calculate the Phi coefficient.
- Re-calculate each of the above coefficients with the following table based on a larger sample.

Husband \ Wife | Clinton | Trump | Other |
---|---|---|---|

Clinton |
24 | 44 | 12 |

Trump |
50 | 42 | 8 |

Other |
6 | 14 | 0 |