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 a two-sided test under a Binomial or HyperGeometric, 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 |
---|---|---|

\(T \sim \mathrm{Bin}(n=9, p=0.5)\) | \(\mathbb{P}(T \leq 4)\) | 0.5 |

\(T \sim \mathrm{Bin}(n=9, p=0.5)\) | \(\mathbb{P}(T \leq 3)\) | 0.25 |

\(T \sim \mathrm{Bin}(n=9, p=0.5)\) | \(\mathbb{P}(T \leq 7)\) | 0.9804688 |

\(T \sim \mathrm{Bin}(n=10, p=0.4)\) | \(\mathbb{P}(T \leq 4)\) | 0.6331033 |

\(T \sim \mathrm{Bin}(n=10, p=0.4)\) | \(\mathbb{P}(T \leq 3)\) | 0.3822806 |

\(T \sim \mathrm{Bin}(n=11, p=0.5)\) | \(\mathbb{P}(T \leq 7)\) | 0.8867188 |

\(T \sim \mathrm{Bin}(n=12, p=0.7)\) | \(\mathbb{P}(T \leq 9)\) | 0.7471847 |

\(T \sim \mathrm{Bin}(n=12, p=0.7)\) | \(\mathbb{P}(T \leq 10)\) | 0.914975 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(\mathbb{P}(T \leq 5 )\) | 0.0206947 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(\mathbb{P}(T \leq 13 )\) | 0.9423409 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(\mathbb{P}(T \leq 6 )\) | 0.0576592 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(\mathbb{P}(T \leq 12 )\) | 0.868412 |

\(T \sim \mathrm{Bin}(n=24, p=0.5)\) | \(\mathbb{P}(T \leq 21)\) | 0.9999821 |

\(T \sim \mathrm{Bin}(n=24, p=0.5)\) | \(\mathbb{P}(T \leq 20)\) | 0.9998614 |

\(T \sim \mathrm{Bin}(n=24, p=0.7)\) | \(\mathbb{P}(T \leq 9)\) | 0.0009834 |

\(T \sim \mathrm{Bin}(n=24, p=0.7)\) | \(\mathbb{P}(T \leq 10)\) | 0.0036332 |

\(T \sim \mathrm{Bin}(n=24, p=0.7)\) | \(\mathbb{P}(T \leq 20)\) | 0.9576025 |

\(T \sim \mathrm{Bin}(n=24, p=0.7)\) | \(\mathbb{P}(T \leq 21)\) | 0.9881259 |

\(T \sim \mathrm{Bin}(n=112, p=0.5)\) | \(\mathbb{P}(T \leq 38)\) | 0.000430 |

\(T \sim \mathrm{Bin}(n=112, p=0.5)\) | \(\mathbb{P}(T \leq 37)\) | 0.0008473 |

\(T \sim \chi^2_1\) | \(\mathbb{P}(T \leq 11.56)\) | 0.9993261 |

\(T \sim \chi^2_1\) | \(\mathbb{P}(T \leq 10.18)\) | 0.9985803 |

\(T \sim \chi^2_1\) | \(\mathbb{P}(T \leq 9.38)\) | 0.9978063 |

\(T \sim \chi^2_1\) | \(\mathbb{P}(T \leq 6.44)\) | 0.9888421 |

\(T \sim \mathrm{HyperGeom}(N=45, r=24, c=19)\) | \(\mathbb{P}(T \leq 15)\) | 0.999568 |

\(T \sim \mathrm{HyperGeom}(N=54, r=45, c=18)\) | \(\mathbb{P}(T \leq 15)\) | 0.639705 |

\(T \sim \mathrm{HyperGeom}(N=55, r=24, c=19)\) | \(\mathbb{P}(T \leq 15)\) | 0.999987 |

\(T \sim \mathrm{HyperGeom}(N=55, r=45, c=18)\) | \(\mathbb{P}(T \leq 15)\) | 0.709572 |

\(T \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(T \leq -1.7973)\) | 0.036144 |

\(T \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(T \leq 1.1034)\) | 0.865073 |

\(T \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(T \leq 1.2695)\) | 0.897869 |

\(T \sim \mathcal{N}(0,1)\) | \(\mathbb{P}(T \leq 1.7973)\) | 0.963856 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.05\) | 6 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.95\) | 14 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.10\) | 7 |

\(T \sim \mathrm{Bin}(n=20, p=0.5)\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.90\) | 13 |

\(T \sim \chi^2_{4}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.90\) | 7.77944 |

\(T \sim \chi^2_{4}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.99\) | 13.2767 |

\(T \sim \chi^2_{8}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.90\) | 13.36157 |

\(T \sim \chi^2_{8}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.99\) | 20.09024 |

\(T \sim \chi^2_{12}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.99\) | 26.21697 |

\(T \sim \chi^2_{12}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.90\) | 18.54935 |

\(T \sim \chi^2_{24}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.99\) | 42.97982 |

\(T \sim \chi^2_{24}\) | \(t\) such that \(\mathbb{P}(T \leq t) = 0.90\) | 33.19624 |

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.

Metro believes that 70% of all passengers that travel between 11pm and 3am have not witnessed a crime while riding on the trains. To check, a short survey was given to the first twelve passengers that said that they travel on Metro between 11pm and 3am on weekends. Of the twelve passengers, 10 of them said that they have not witnessed a crime on the train. This is evidence supporting that the claimed proportion may be different, but is this statistically significant evidence to conclude this at the five percent level of significance?

The time interval between eruptions of Old Faithful geyser is recorded 112 times to see whether the median interval is less than or equal to 60 minutes or not (i.e., 60 minutes is above the median). Of the 112 intervals, 38 are 60 minutes or less. What do you conclude about the median time interval?

An item A is manufactured using a certain process. Item B serves the same function as A but is manufactured using a new process. The manufacturer wishes to determine whether B is preferred to A by the consumer, so she selects a random sample consisting of 10 consumers, gives each of them one A and one B, and asks them to use the items for some period of time. At the end of the allotted period of time the consumers report their preferences to the manufacturer. Eight consumers preferred B to A, one preferred A to B, and 1 reported “no preference”. Is there enough evidence to conclude that customers prefer item B to item A?

Prior to a nationally televised debate between two presidential candidates, a random sample of 100 persons stated their choice of candidates as follows. Eighty-four persons favored the Democratic candidate, and the remaining 16 favored the Republican. After the debate the same 100 people expressed their preference again. Of the persons who formerly favored the Democrat, exactly one-fourth of them changed their minds, and also one-fourth of the people formerly favoring the Republican switched to the Democratic side. Was the voting alignment in the population altered by the debate?

The total annual precipitation is recorded yearly for 19 years. The precipitation in inches was 45.25, 45.83, 41.77, 36.26, 45.37, 52.25, 35.37, 57.16, 35.37, 58.32, 41.05, 33.72, 45.73, 37.90, 41.72, 36.07, 49.83, 36.24, and 39.30. Is precipitation tending to increase or decrease, or is it staying about the same?

Two carloads of manufactured items are sampled randomly to see how many items they were carrying were defective. From the first carload, 13 of the 86 items were defective. From the second carload 17 of the 74 items were considered defective. Is the proportion different for the two carloads?

The eastern chipmunk trills when pursued by a predator, possibly to warn other chipmunks. Researchers released chipmunks either 10 or 100 meters from their home burrow, then chased them (to simulate predator pursuit). Out of 24 chipmunks released 10 meters from their burrow, 16 trilled and 8 did not trill. When released 100 meters from their burrow, only 3 chipmunks trilled, while 18 did not trill. Do chipmunks trill more when they are close to home?

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.

A random sample of tenth-grade boys resulted in the following 20 observed weights: 81, 86, 93, 98, 103, 103, 117, 119, 119, 122, 128, 131, 134, 137, 142, 144, 154, 158, 161, 165. Find an approximate 90% confidence interval for the median, and provide the exact confidence coefficient.

An engineer writing the acceptance specs for a load of steel reinforcement rods would like to specify that at least 90% of the rods are between the sixth longest and the sixth shortest rods in a random sample she selects. In order to have 99% confidence in this statement, what should the sample size be?

A fitness gym has measured the percentage of fat on 86 of its members. At least what percent of its members have a fat percentage between \(X^{(2)}\) and \(X^{(85)}\) with 90% certainty.