MAT MILESTONE 2

Dataset Description

File: MAT 202 YourPace Milestone 2 Data

The dataset contains 500 observations with the following variables:

· ComponentID: Unique identifier for each component

· ProcessingTime: Time (in minutes) required to machine each component

· WaitTime: Time (in minutes) customers wait for technical support

· FailureTime: Time (in hours) until component failure in stress testing

Part A: Continuous Probability Density Functions

Background

You need to understand the fundamental properties of continuous probability distributions to properly analyze TechFlow's quality metrics.

A1. Theoretical Foundation

Using the Processing Time data from your dataset:

a) Import the ‘MAT 202 YourPace Milestone 2 Data’ file into Minitab and create a histogram of the Processing Time variable. Based on the shape of the distribution, identify what type of continuous distribution this appears to follow.

b) Calculate the sample mean and standard deviation of the processing time using Minitab's descriptive statistics function.

c) For any continuous probability distribution, explain why P(X = k) = 0 for any specific value k. Demonstrate this concept using your processing time data by calculating P(X = 30.0000) exactly.

Part B: Uniform Probability Distributions

Background

TechFlow's customer service department has found that customer wait times follow a uniform distribution. You need to analyze this distribution to improve service efficiency.

B1. Distribution Identification and Parameters

a) Using Minitab, create a histogram of the ‘Wait Time’ variable with appropriate bin settings. Verify that this data appears to follow a uniform distribution.

b) From your sample data, estimate the parameters a and b for the uniform distribution U(a,b). Use Minitab to find the minimum and maximum values of wait time.

c) Write the probability density function f(x) for this uniform distribution, including the range of x values.

B2. Probability Calculations

d) Calculate the mean and standard deviation for this uniform distribution using the Minitab.

e) Using the uniform distribution properties, calculate:

· P(Wait Time < 8 minutes)

· P(Wait Time > 15 minutes)

· P(5 < Wait Time < 12 minutes)

B3. Percentile Analysis

f) Management wants to ensure that 90% of customers wait less than a certain time. What should this target time be?

B4. Conditional Probability

g) A customer has already been waiting for 10 minutes. What is the probability they will wait more than 15 minutes total? Use the conditional probability approach for uniform distributions presented in the provided materials for this section as “the first way” approach.

Part C: Exponential Probability Distributions

Background

Component failure times in stress testing follow an exponential distribution. This analysis is crucial for determining warranty periods and reliability standards.

C1. Distribution Verification and Parameters

a) Using Minitab, create a histogram of the Failure Time variable. Does this appear to follow an exponential distribution? What characteristics support this conclusion?

b) Calculate the sample mean of failure times using Minitab. For an exponential distribution, this sample mean estimates the parameter μ (mean time between failures).

c) Calculate the decay parameter m = 1/μ for this exponential distribution.

C2. Reliability Analysis

b) Using the exponential distribution properties, calculate:

· P(Failure Time > 20 hours)

· P(Failure Time < 10 hours)

· P(15 < Failure Time < 30 hours)

c) Find the median failure time (50th percentile) using Minitab's percentile function.

d) Management wants to set a warranty period such that only 10% of components fail within the warranty period. What should this warranty period be?

C3. Memoryless Property

e) A component has already been operating successfully for 20 hours. What is the probability it will operate for at least another 10 hours?