# Mathematics

1. Redundancy in Aircraft Radios: The FAA requires that commercial aircraft used for flying in instrument conditions must have two independent radios instead of one. Assume that for a typical flight, the probability of a radio failure is 0.0035. What is the probability that a particular flight will be safe with at least one working radio? Why does the usual rounding rule of three significant digits not work here? Is this probability high enough to ensure flight safety?

2. Illinois Lottery:

a. In the Illinois Little Lotto game, you win the jackpot by selecting five different whole numbers from 1 through 39 and getting the same five numbers (in any order) that are later drawn. What is the probability of winning a jackpot in this game?

b. In the Illinois Pick 3 game, you win a bet by selecting three digits (with repetition allowed) and getting the same three digits in the exact same order as they are later drawn. What is the probability of winning this game?

c. The Illinois Pick 3 game returns $500 for a winning $1 ticket. What should be the return if Illinois were to run this game for no profit?

3. Mendelian Genetics: When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 580 peas, and Mendel theorized that 25% of them would be yellow peas.

a. If Mendel’s theory is correct, find the mean and standard deviation for the numbers of yellow peas in such groups of 580 offspring peas.

b. The actual results consisted of 152 yellow peas. Is that result unusually high? What does this result suggest about Mendel’s theory?

4. Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. Draw a graph in each case. Find the probability that a randomly selected adult has an IQ between 110 and 120 (referred to as bright normal).

Using the Central Limit Theorem, use this information about the overhead reach distances of adult females: μ = 205.5 cm , σ = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stations:

a. If 1 adult female is randomly selected, find the probability that her overhead reach is between 180.0 cm and 200.0 cm.

b. If 50 adult females are randomly selected, find the probability that they have a mean overhead reach between 198.0 cm and 206.0 cm.

5. Use the given data to find the minimum sample size required to estimate a population proportion or percentage. Fortune Tellers Find the sample size needed to estimate the percentage of adults who have consulted fortune tellers. Use a 0.03 margin of error, use a confidence level of 98%, and use results from a prior Pew Research Center poll suggesting that 15% of adults have consulted fortune tellers.

6. Assume a significance level of α = 0.05 and use the given information for the following:

a. State a conclusion about the null hypothesis. (Reject H 0 or fail to reject H 0 . )

b. Without using technical terms, state a final conclusion that addresses the original claim.Original claim: Women have heights with a standard deviation equal to 5.00 cm. The hypothesis test results in a P-value of 0.0055.

7. Identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution. Assume a 0.05 significance level and answer the following:

a. Is the test two-tailed, left-tailed, or right-tailed?

b. What is the test statistic?

c. What is the P-value?

d. What is the null hypothesis, and what do you conclude about it?

e. What is the final conclusion?

Guns in the Home: In a Gallup poll of 1003 randomly selected subjects, 373 said that they have a gun in their home. The accompanying Minitab display shows results from a test of the claim that 35% of homes have guns in them.

MINITAB

picture of minitab: Test of p = 0.35 vs p not -0.35

Variable X N sample p 95% CI Z-Value P- Value

Guns 373 1003 0.371884 (0.341974, 0.401795) 1.45 0.146

Testing Hypotheses. Assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), critical value(s), and state the final conclusion that addresses the original claim.

20. Ages of Race Car Drivers Listed below are the ages (years) of randomly selected race car drivers (based on data reported in USA Today). Use a 0.05 significance level to test the claim that the mean age of all race car drivers is greater than 30 years.