1. [From Mike] The distribution of a discrete random variable is
a. Always normal.
b. A probability distribution described by a density curve.
c. A histogram with the mean at the balance point.
d. A curve that may or may not have a normal shape.
2. A statistician has been taking simple random samples of size n from a population. She discovers that the standard deviation of the sampling distribution of the sample means is too large for the goals of the study. To cut the size of the standard deviation of the sampling distribution of the sample means in half she must:
a) Double the size of the random samples to 2n
b) Quadruple the size of the random samples to 4n
c) Halve the size of the random samples to 0.5n
d) Quarter the size of the random samples to 0.25n
3. Circle the best answer. An enormous vat of red and blue marbles is 50% red. As more and more marbles are pulled from the vat:
a. The percentage of blue marbles pulled from the vat tends to get closer to 50% and the number of red and blue marbles pulled from the vat start to even out.
b. The percentage of blue marbles pulled from the vat tends to get closer to 50% and the number of red marbles pulled from the vat tends to move further away from half of the marbles.
c. The percentage of blue marbles pulled from the vat tends to move further away from 50% and the number of red and blue marbles pulled from the vat start to even out.
d. The percentage of blue marbles pulled from the vat tends to move further away from 50% and the number of red marbles pulled from the vat tends to move further away from half of the marbles.
4. [Based on a question from Leah] In a game of chance, you win if a roll of a die is either a 1 or a 6. You decide to roll the die 45 times. X is the number of times you win.
a. Is this a binomial setting? Support your answer.
[Jamie's answer] Yes, it is a binomial setting. There are a fixed number of observations (45). The observations are independent from each other (knowing if one roll is a success has not effect on future rolls). Each observation is either a success (1,6) or a failure (2,3,4,5,6) and the probability of a success is the same on each roll (2/6).
b. Showing your work, determine the probability of winning EXACTLY 13 times.
c. Approximate the probability of winning 13 or more times. Show your work.
Use the normal distribution to approximate the binomial distribution
Find P(X>13) by finding shaded area for the N(15, 10) by
Probability of winning 13 or more times is 0.736
d. Is this a random phenomenon? If not explain why. If it is, explain why and include a sketch of part or all of a plot labeled with some specific values (numbers) which supports your definition of random phenomenon.
[Lily's answer] A random phenomenon is one whose individual outcomes are uncertain but nevertheless a regular distribution (pattern) of outcomes emerges over a long number of repetitions. This is a random phenomenon because the outcome of one roll is uncertain but the following distribution does emerges [which is exactly described by the binomial probability formal and approximately described by N(15,3.16)
5. A machine which makes short pipes is checked by computing the mean of diameters of samples of 10 pipes. Below is a process control chart shows 10 such samples.
a. [Based on a question from Chris] Sample six is above the control limit. At the time this point was plotted, the operator did not know what the remainder of the control chart would look like. However, he does understand statistics. He argued statistically that the process generating the control chart may not need to be corrected? What was his argument?
[Mike's answer] His argument was that this point was the result of chance.... According to the 68-95-99.7 rule, 0.3% of the data falls outside the control limits so this may just be one of those random chances, rather than an error in the process.
b. [Based on a question from Mike] Explain why sample means are often used to construct control chart.
[Brad's Answer] Because of the CLT even non-normal distributions which have sample-mean distributions which are approximately normal.
c. If the control limits shown are determined from all 100 pieces of data used to create the chart, compute the standard deviation of these 100 individual data points.
6. The average number of books in the homes of all BB&N students is 1000. You have selected 25 homes at random and the first two you look at have 900 books and 950 books respectively. What do you expect the mean number of books to be for the entire sample (numerical answer).
7. a. State the CLT for Sample Means.
Refer to your book.
b. Make up and SOLVE any simple problem which involves using the TI-83 command normalcdf which requires the use of the CLT for Sample Means.
Rachel's answer] The average grade in math for all Massachusetts high school sophomores is 3.1 (about a B). The standard deviation if 0.798. A sample [SRS] of 100 sophomores is taken . What is the probability that the sample mean is great than 3.0?
8. Twenty percent of all items produced on an assembly line require repainting. For a random sample of 100 items, the probability that at least "k" percent of all items produced on an assembly line require repainting is 0.1056. Including a fully annotated presentation with rough plot, determine k.