SMMD: Practice Problem Set 2 Topics: Type I and Type II errors, Sampling Distributions and Central Limit Theorem

1. Recall the example we discussed in class on “Decision Making in Uncertain

Situations”. The weights of the cereal boxes follow a normal distribution with mean 16.3 oz and std dev 0.2 oz. Suppose the manager sets a threshold at µ - 2 σ, i.e. 16.3 – 2*0.2 = 15.9 oz., and decides to stop the packaging process and conduct an investigation if a cereal box with weight lower than 15.9 oz is observed. What is the probability of Type I error with this decision rule?

2. In the above question, rather than raise an alarm when a box with weight lower than the threshold weight is observed, the manager has decided to wait until 2 consecutive boxes have weights lower than the threshold. What is the probability of a Type I error if this procedure is used? Assume that each weight of each cereal box is independent of the other. Has the manager increased or decreased the chance for a Type I error compared to if she raised an alarm after only one observation? How about the chance for a Type II error?

3. For each of the following statements, indicate whether it is True/False. If false, explain why.

The manager of a warehouse monitors the volume of shipments made by the delivery team. The automated tracking system tracks every package as it moves through the facility. A sample of 25 packages is selected and weighed every day. Based on current contracts with customers, the weights should have µ = 22 lbs. and σ = 5 lbs.

I. Before using a normal model for the sampling distribution of the average package weights, the manager must confirm that weights of individual packages are normally distributed.

II. The standard error of the daily average SE( ) = 1.

4. An educational startup that helps MBA aspirants write their essays is targeting individuals who have taken GMAT in 2012 and have expressed interest in applying to FT top 20 b-schools. There are 40000 such individuals with an average GMAT score of 720 and a standard deviation of 120. The scores are distributed between 650 and 790 with a very long and thin tail towards the higher end resulting in substantial skewness. Which of the following is likely to be true for randomly chosen samples of aspirants?

A. The standard deviation of the scores within any sample will be 120. B. The standard deviation of the mean of across several samples will be 120. C. The mean score in any sample will be 720.

D. The average of the mean across several samples will be 720. 5. At the beginning of your stay at ISB, you started a YouTube channel to host videos of

your b-school life so that your friends and family can remain connected with you. You posted a total of 240 videos, roughly one for every working day. You are analyzing the data dump available from Google Analytics at the end of the year. Among your 500 Facebook friends, who subscribed to your channel, the average number of views is 25 with a standard deviation of 50. Google Analytics allows you to choose a random sample of 25 friends and send them an online gaming credit for every view of your channel as a token of appreciation. What is the approximate probability that you will shell out less than 20 credits?

A. 30% B. 6% C. 0% D. 1.25% E. Cannot determine because standard deviation is greater than the mean

6. Auditors at a small community bank randomly sample 100 withdrawal transactions

made daily at an ATM machine located near the bank’s main branch. Over the past 2 years, the average withdrawal amount has been $50 with a standard deviation of $40. Since audit investigations are typically expensive, the auditors decide to not initiate further investigations if the mean transaction amount of the sample is between $45 and $55. What is the probability that in any given week, there will be an investigation?

A. 1.25% B. 2.5% C. 10.55% D. 21.1% E. 50%

7. The auditors from the above example would like to maintain the probability of

investigation to 5%. Which of the following represents the minimum number transactions that they should sample if they do not want to change the thresholds of 45 and 55?

A. 144 B. 150 C. 196 D. 250 E. Not enough information

8. In question 6, suppose that the mean transaction amount of the sample is found to be

$70, and consequently an investigation is held. However, no reasons for the increase in withdrawal size are uncovered. This means that:

A. The auditors commit a Type I error B. The auditors commit a Type II error C. Since no problems are uncovered, there is no error D. Since the mean transaction amount is higher than $55, there is no error E. There is no enough information to ascertain whether an error was

committed

9. In the scenario in Question 6, an upcoming holiday weekend increases the mean size of withdrawals. If the sample average on such a day remains inside the thresholds of $45 and $55 (also called control limits), then the auditors do not conduct an investigation. Thus, a:

A. a Type I error has occurred B. a Type II error has occurred C. Neither Type I nor Type II error has occurred D. Both Type I and Type II errors have occurred

10. When everything is proceeding normally, a process produces items having mean weight of 100 units and s.d. 10 units. The process is considered to be going normally when the items produced are neither too light, nor too heavy. Since the distribution of the weights produced by the process is not known, the manager decides to take a randomly selected sample of items and check whether the mean weight of the sample is too high, or too low. The manager knows that the distribution of the mean weights can be taken to be normal due to Central Limit Theorem.

Because the manager wants to check process deviations on both sides of the mean, she must set a lower and an upper threshold. For a type I error probability of 0.27%, determine upper and lower thresholds for the sample mean for each of the following sample sizes. Assume that conditions for applying CLT are satisfied. Note: These lower and upper thresholds are also called the Lower Control Limit (LCL) and the Upper Control Limit (UCL), respectively

I. 4 II. 5 III. 6 IV. 10