HOMEWORK -

Review Questions - The following 8 questions are meant to review the previous material just in case you are still having difficulties.

Q1. Suppose that Z ∼ N(0,1) and X ∼ N(µ = 2,σ = 2). Answer the following questions using R or Normal Tables (or Crunch It! if you prefer). When you answer the questions provide a sketch of the normal distribution of copy and paste an annotated sketch of the normal distribution. Understanding the bell curve is so important and you must know this distribution inside and out. It helps to draw and shade in areas when you solve problems at first.

(a) P(X < 3.3)

(b) P(Z > -1.8)

(c) P(X < 1.4)

(d) P(Z = 1.3)

(e) P(1.3 < X < 2.8)

(f) Find P((X > 4)∪(X < 1))

(g) Find P(|Z| > 2)

(h) Find P(|X -2| > 1.5)

Q2. Now the following problems are quantile problems which ask for values of x on the x-axis for which certain probabilities are true. Make sure you draw bell curves and shade in regions just like the procedure I showed you in class to solve the following problems. Suppose that Z ∼ N(0,1) and X ∼ N(µ = 3,σ = 3).

(a) Find the value of a such that P(Z < a) = 0.32.

(b) Find the value of b such that P(X < b) = 0.65.

(c) Find the 50^th percentile of X or the value b such that P(X < b) = 0.5.

(d) Find the 80^th percentile of Z.

(e) Find the value c such that P(Z > c) = 0.2.

(f) Find the value of b such that P(|Z| < b) = 0.9.

(g) Find the value c such that P((X < 2)∪(X > c)) = 0.95.

Q3. Weights of female cats of a certain breed are normally distributed with mean 4.1 kg and standard deviation 0.6 kg.

(a) What proportion of female cats have weights between 3.7 and 4.4 kg?

(b) How heavy is a female cat whose weight is on the 80th percentile?

(c) A female cat is chosen at random. What is the probability that she weighs more than 4.5 kg?

(d) A certain female cat has a weight that is 0.5 standard deviations above the mean. What proportion of female cats are heavier than this one?

Q4. A geneticist is studying two genes. Each gene can be either dominant or recessive. A sample of 100 individuals is categorized as follows.

 (a) What is the probability that a randomly selected individual has a dominant Gene 1?

(b) Given that someone has a dominant Gene 1, what is the probability that person has a dominant Gene 2?

(c) Is the probability of having a dominant Gene 2, independent of having a dominant Gene 1?

(d) What is the probability that someone in this study has a dominant Gene of any kind?

Q5. A population of 600 arterial stents recently arrived from 3 different manufacturers. The stents are categorized by which manufacturer {A, B, C} and whether they conform to a thickness specification.

The following table presents the number of stents in each category. A stent is chosen at random from the population.

(Hint: You should probably add a row and column to the above table.)

(a) If the stent is from manufacturer A, what is the probability that it is conforming?

(b) If the stent is conforming, what is the probability that is is from manufacturer A?

(c) If the stent is conforming, what is the probability that it is not from manufacturer C?

(d) If the stent is not from manufacturer C, what is the probability that it is conforming?

Q6. The proportion of people in a given community who have a certain disease is 0.005. A test is available to diagnose the disease. If a person has the disease, the probability that the test will produce a positive signal is 0.99. If the person does not have the disease, the probability that the test will produce a positive signal is 0.01.

(a) Construct a complete tree diagram for all possibilities of testing and disease status.

(b) What is the probability that a person will test positive?

(c) Given that a person tests positive, what is the probability that the person actually has the disease?

Q7. Freckles and moles on humans are very common. However usually large, abnormally shaped and/or abnormally colored freckles/moles can sometimes be warning signs of skin cancer. Let X be the number of "unusual" freckles/moles found on patients that are seen by a dermatologist's office in Bloomington. We'll assume that an individual can only have between 0 and 4 unusual moles/freckles each, hence X = {0,...,4}. The population probability distribution for the number of unusual freckles/moles is presented in the following table.

(a) Find P(X ≤ 2).

(b) Find P(X > 1).

(c) Find P[X < 2 or X ≥ 3].

(d) Find µ_X.

(e) Find σ²_X.

Q8. If X and Y are independent random variables with means µ_X = 9.5 and µ_Y = 6.8 and standard deviations σ_X = 0.4 and σ_Y = 0.1, find the means and standard deviations of the following:

(a) W = 3_Y -2_X, hence find µW and σ_W.

(b) Z = X +4Y -0.2, hence find µ_Z and σ_Z.

New Questions - The following 3 questions will cover some of the new material on confidence intervals for the mean.

Q9. The table below contains 5 random samples of size 10 from the population of infants who were born at a hospital in New York. The birth weights in (oz) appear in the table below.

For each sample, you can quickly calculate sample mean and standard deviation by loading in the data into R in the form of a vector and then using the functions mean() and sd() to calculate x¯ and S for the sample. For example for the first sample I can use the commands in R as follows to load in the data into a vector x and then calculate the mean x¯ and standard deviation S

x= c(97,117,140,78,98,148,108,135,126,121)

xbar = mean(x)

s = sd(x)

print(xbar)

print(s)

(a) Calculate the sample mean x¯ and the sample standard deviation s for the first sample.

(b) Assume the population standard deviation is σ = 22. For the first sample, construct a 95% confidence interval for the population mean µ based upon the central limit theorem and the normal distribution.

(c) For the first sample, construct a 99% confidence interval for the population mean µ. Which interval is wider, the 95% interval or the 99% interval?

(d) Assuming the population standard deviation is σ = 22. For the second sample, construct a 95% confidence interval for the population mean µ based upon the central limit theorem and the normal distribution. How does the 95% confidence interval for the second sample compare with the 95% confidence interval for the first sample in terms of width and position?

(e) For the 3rd, 4th and 5th samples construct the 95% confidence intervals assuming σ = 22. Why do you get different answers for different samples?

(f) In the previous problems we assumed that we know what σ was. Now let us drop that assumption. Calculate the 95% confidence interval for the first and second sample using the T distribution confidence interval formula.

Q10. An important indicator of lung function is forced expiratory volume (FEV), which is the volume of air that a person can expire in one second. Dr. Hernandez plans to measure FEV in a random sample of n young women from a certain population, and to use the sample mean y as an estimate of the population mean. Let E be the event that Hernandez's sample mean will be within ±100 ml of the population mean. Assume that the population distribution is normal with mean µ = 3000 ml and standard deviation σ = 400 ml. Find Pr{E} if

(a) n = 15

(b) n = 60

(c) How does Pr{E} depend on sample size?

Q11. At a large university, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn. What is the probability that the average age of these students is greater than 23 years? (Hint: The sample size is large and you know σ, so you can use the central limit theorem.)

Please use R program.