Homework 10

You should turn in a hand-written copy of your solutions to these problems at the beginning of Lecture 12, showing all relevant work.

Critical Values and Margins of Error

For each of the following problems:

Determine the body probability $c$ and overall-tail probability $\alpha$.
Determine the appropriate critical value $z_{\alpha/2}$.
Determine the probability $c$ margin of error $E$ for the sample mean $\bar{X}$.

You may (and should) use R to compute all of these values. However, also show your work in your hand-written solutions.

A random sample of size 10 is collected from a Normal population with mean $\mu_{X} = 10$ and standard deviation $\sigma_{X} = 100$. A 99% margin of error for the sample mean is desired.
A random sample of size 75 is collected from a Normal population with mean $\mu_{X} = -10$ and standard deviation $\sigma_{X} = 2$. A 50% margin of error for the sample mean is desired.
A random sample of size 5 is collected from a Normal population with mean $\mu_{X} = 100$ and standard deviation $\sigma_{X} = 70$. A 75% margin of error for the sample mean is desired.

For each of the following problems:

Determine the population under consideration.
Determine the sample from the population.
Determine the population mean.
Identify the sample statistics of the sample, including the sample size, sample mean, and sample standard deviation.
Compute the estimate of the standard error of the sample mean.
Compute the $T$-score for the sample mean.
Compute the probability that a sample mean from such a random sample would exceed the observed sample mean of the collected sample, if we assume that both: (1) the distribution of the variable in the population is Normal and (2) the standard deviation of the population equals the standard deviation of the sample.

You may (and should) use R to compute all of these values. However, also show your work in your hand-written solutions.

An agricultural researcher measured the yields of head corn in pounds per acre in 10 randomly selected fields of corn in England. She found that the average yield was 117.2 pounds per acre, and the standard deviation across the fields was 12.1 pounds per acre. Assume the average yield in the population is 115 pounds per acre.
Monmouth Medical Center conducted a study to estimate hospital costs for accident victims who wore seatbelts. Twenty randomly selected cases from the hospital's records have a histogram that appears bell-shaped, and the mean and standard deviation of the twenty values is $9004 and $5629, respectively. Assume the mean cost of accidents in the population is $8500.
A physician wants to develop criteria for determining whether a senior citizen's pulse rate is atypical. She collects a random sample of 40 senior citizens from a nursing home. The average and standard deviation of the pulse rates in the sample was 69.4 beats per minute and 11.3 beats per minute, respectively. Assume the mean pulse rate in the population is 70.
An archeologist collected a random sample of 12 male Egyptian skulls from an archeological dig dating from 4000 BCE. The mean maximum breadth of the sampled skulls was 128.7 cm, and the standard deviation across the skull maximum breadths was 4.63 cm. Assume the mean maximum breadth of skulls in the population is 125 cm.