Department of Statistics, Virginia Tech

This homework is due on **Tuesday, September 26th at 2pm** (the start of class). Please turn in all your work. This homework primarily quantile tests and tolerance limits.

**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.

It is known that 20% of a certain species of insect exhibit a particular characteristic A. Eighteen insects of that species are obtained from an unusual environment, and none of these have characteristic A. Is it reasonable to assume that insects from that environment have the same probability of 0.20 of having that characteristic, as the species in general has?

- (10 pts) Perform the test “by hand”.
*(See note about about what that means.)* - (5 pts) Perform the test “using a software library”.
*(Again, see above.)*

In a dice game a pair of dice were thrown 180 times. The event “seven” occurred on 38 of those times. If these dice are fair the probability of “seven” is one-sixth. If they are loaded the probability is higher. Is the probability of “seven” what should it be if the dice were fair? Use the appropriate one-tailed test.

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

A random sample of tenth-grade boys resulted in the following 20 observed weights.

`weight <- c(142, 134, 98, 119, 131, 103, 154, 122, 93, 137, 86, 119, 161, 144, 158, 165, 81, 117, 128, 103)`

- Test the hypothesis that the median weight is 103.
- (10 pts) Perform the test “by hand”.
- (5 pts) Perform the test “using a software library”.

- Test the hypothesis that the third decile is no greater than 100. A decile is any of the nine values that divide the sorted data into ten equal parts, so that each part represents 1/10 of the sample population.
- (10 pts) Perform the test “by hand”.
- (5 pts) Perform the test “using a software library”.

It is desired to design a given automobile to allow enough headroom to accommodate comfortably all but the tallest 5% of the people who drive. Former studies indicate that the 95th percentile was 70.3 inches. In order to see if the former studies are still valid, a random sample of size 100 is selected. It is found that the 12 tallest persons in the sample have the following heights.

`heights <- c(72.6, 70.0, 71.3, 70.5, 70.8, 76.0, 70.1, 72.5, 71.1, 70.6, 71.9, 72.8)`

Is it reasonable to use 70.3 as the 95th percentile?

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

- (10 pts) What must the sample size be to be 90% sure that at least 95% of the population lies within the sample range?
- Use the exact calculation.
- Use the approximation.

- (10 pts) A fitness gym has measured the percentage of fat in 86 of its members.
- At least what percent of its members have fat percentages between the smallest and the largest of the percentages measured on the 86 members in the sample, with 95% certainty?
- At least what percent of its members have fat percentage between \(X^{(2)}\) and \(X^{(85)}\) with 95% certainty?

- (5 pts) 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?