OMGT 6123 Quantitative Methods

Exercise #2

Probability and Sampling

Download the Excel file titled Exercise 2 for this assignment. The file contains several tabs with the data required for the assignment.

Problem 1 (10 points)

In this problem, we will analyze basic probability concepts. For each calculation provide your answer and explain how you made the calculation. You may show a formula, an Excel formula, a table, or describe textually. Your goal is to explain your approach clearly and succinctly. Do the following:

Recently Apple announced the new iPhone 11 Pro, and iPhone 11 Pro Max. They are offered with a number of options including:

o Four Colors: Space Grey, Silver, Midnight Green and Gold

o Three Capacities: 64 GB, 256GB and 512 GB

o 4 Carriers (AT&T, Sprint, Verizon, T-Mobile) and carrier free

At the same time, they announced the iPhone 11 with the following options:

o Six Colors: Black, Green, Yellow, Purple, Red and White

o Three Capacities: 64 GB, 256GB and 512 GB

o 4 Carriers and carrier free

Based on this information calculate the total number of new configurations available.

Calculate the total number of configurations available ignoring the carrier.

A retailer wants to display 5 phones in an interactive display.

o Calculate the total number of ways the retailer can select phone configurations for the display, ignoring carrier and order.

o Recalculate the number of ways the phones can be displayed if order of display matters.

Assume that a battery defect effects 1% of the phones. Calculate the following probabilities for the 5 display phones

o None of the phones have the defect

o Exactly one of the phones has the defect.

o The set of 5 phones has 1 or less defects.

Now assume the retailer wants to stock 10 phones of each configuration for the no carrier option and 25 for each of the 4 carriers. Based on this population, calculate the following:

o The total number of phones to stock

o The probability a randomly selected iPhone is Silver.

o The probability a randomly selected iPhone is Red.

o The probability a randomly selected iPhone is an iPhone 11 Pro given that it is Gold.

o The probability a randomly selected iPhone is an iPhone 11 given that it is has a 256 GB capacity.

Problem 2 (10 points)

In this problem, we will analyze discrete probability distributions.

First consider the following probability distribution for the total number of devices that connect to a home router.

XP(X)

16%

24%

32%

43%

5 4%

64%

74%

85%

98%

1010%

119%

128%

138%

146%

155%

164%

173%

183%

192%

202%

Generate plots of the PDF and CDF for this distribution.

Calculate the following

o The expectation and the variance of the number of devices.

o The probability that a customer has 10 or fewer devices connecting.

o The probability that a customer has 14 or more devices connecting.

o The 50th percentile of the distribution.

Now assume that the amount of data transferred per day is normally distributed with a mean of 15 GB and a standard deviation of 5 GB. Assume the data transferred each day is independent of other days. Calculate the following

o The probability a customer transfers more than 18 GB in a day.

o The probability a customer transfers more than 18 GB for 3 days in a row.

o The 90th percentile of the daily transfer amount.

Data for Problem 3

Problem 3 (10 points)

In this problem we will analyze the Corolla Car data provided in the HW Data file.

Calculate the mean, median, and standard deviation of o Price

o Age_08_04

o KM

Use Excel's Data Analysis Add-in to generate a random sample of 25 cars. Based on the sample calculate the mean, median, and standard deviation of o Price

o Age_08_04

o KM

Use XLMiner or R to generate a random sample of 25 cars. If you use XLMiner, paste the summary tables (labeled Inputs) into your template. If you use R, paste your sample statement into the template. Based on the sample calculate the mean, median, and standard deviation of o Price

o Age_08_04

o KM

Use XLMiner or R to generate a stratified sample of 25 cars using Mfg_Year as the Strata variable with samples proportional to strata size. If you use XLMiner, paste the summary table (labeled Inputs) into your template. If you use R, paste your sample statement into the template. Based on the sample calculate the mean, median, and standard deviation of o Price

o Age_08_04

o KM

Generate a box plot that shows the distribution of the price from the original prices and the 3 samples side by side.