# Mathematics

Spell-checking software catches “nonword errors,” which are strings of letters that are not words, as when “the” is typed as “eth.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number *X* of nonword errors has the following distribution.

Value of X |
0 | 1 | 2 | 3 | 4 |

Probability | 0.06 | 0.2 | 0.32 | 0.32 | 0.10 |

(a) Sketch the probability distribution for this random variable.

(b) Write the event “at least one nonword error” in terms of *X*.

*X* ≥ 2

*X* > 1

*X* ≥ 1

*X* ≤ 1

What is the probability of this event? (c) Describe the event

*X* ≤ 2

in words.

no more than three nonword errorsless than 2 nonword errors more than 2 nonword errorsno more than two nonword errors

What is its probability? What is the probability that

*X* < 2?

2. Sheila’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila’s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with *μ* = 128 mg/dl and *σ* = 10 mg/dl. What is the level *L* such that there is probability only 0.05 that the mean glucose level of 2 test results falls above *L* for Sheila’s glucose level distribution? (Round your answer to one decimal place.) mg/dl

3. It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as *Benford’s Law*, shown below.

First digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Proportion | 0.289 | 0.187 | 0.14 | 0.081 | 0.057 | 0.07 | 0.05 | 0.037 | 0.089 |

Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford’s Law holds, that among 1234 randomly chosen invoices there are no more than 678 in amounts with first digit 1, 2, or 3? (Round your answer to four decimal places.)

4. The weight of the eggs produced by a certain breed of hen is normally distributed with mean 67.2 grams (g) and standard deviation 6.9 g. If cartons of such eggs can be considered to be SRSs of size 12 from the population of all eggs, what is the probability that the weight of a carton falls between 775 g and 825 g? (Round your answer to four decimal places.)

5. The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 65% males and 35% females. The agency calls 30 people chosen at random from its list.

(a) What is the probability that 19 of the 30 people are men? (Use the binomial probability formula. Round your answer to four decimal places.) (b) What is the probability that the first woman is reached on the 5th call? (That is, the first 5 calls give MMMMF. Round your answer to four decimal places.)