91影视

Use the following information to answer the next five exercises. A study was conducted to test the effectiveness of a software patch in reducing system failures over a six-month period. Results for randomly selected installations are shown in Table 10.21. The 鈥渂efore鈥� value is matched to an 鈥渁fter鈥� value, and the differences are calculated. The differences have a normal distribution. Test at the 1% significance level. $$ \begin{array}{|l|l|l|l|l|l|l|}\hline \text { Installation } & {\mathbf{A}} & {\mathbf{B}} & {\mathbf{C}} & {\mathbf{D}} & {\mathbf{E}} & {\mathbf{F}} & {\mathbf{G}} & {\mathbf{H}} \\ \hline \text { Before } & {3} & {6} & {4} & {2} & {5} & {8} & {2} & {6} \\ \hline \text { After } & {1} & {5} & {2} & {0} & {1} & {0} & {2} & {2} \\ \hline\end{array} $$ What is the random variable?

Step by step solution

01

Understand the Context

A study is conducted to test the effectiveness of a software patch. It measures system failures before and after applying the software patch to several installations. Results show the number of failures before and after the patch.

02

Identify Pairwise Data

For each installation (A through H), the number of failures is recorded before and after the patch. This creates pairs of data points for the 'before' and 'after' conditions for each installation.

03

Define the Difference

The difference for each installation is calculated as the number of failures 'before' minus the number of failures 'after'. This difference measures the change in system failures due to the patch.

04

Identify the Random Variable

The random variable is typically the measurement or outcome being analyzed for variation or change. Here, it is the 'difference in number of system failures between before and after applying the patch for each installation.' These differences are assumed to follow a normal distribution for the test.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Paired Sample

When dealing with data that comes in pairs, especially when analyzing the effectiveness of treatments or conditions, we use a paired sample approach. This method is particularly useful because it can effectively control for variability between subjects that is not related to the treatment or condition.
In our exercise, each installation is treated as a subject, and we collect data in pairs: once before the software patch is applied, and once after. By comparing these paired measurements, we can directly assess the impact of the software patch on each installation without the interference of other variables.

• Poor performance before the patch vs after in multiple installations
• Directly correlating system failures in tangible way
• More reliable conclusions by focusing on differences within the same subject

Normal Distribution

Understanding the normal distribution is crucial when performing hypothesis testing, particularly with paired samples. A normal distribution is a bell-shaped curve that is symmetrical about the mean. Most data points lie close to the mean, with fewer points appearing as you move further away from it. In the context of our study, we assume that the differences in system failures before and after applying the software patch follow a normal distribution. This assumption allows us to use statistical methods that require normally distributed data to draw conclusions about the treatment's effectiveness. However, keep in mind:

• Sufficient sample size helps ensure normal distribution
• Normalized data relies on the assumption that external factors minimally affect results
• Helps in utilizing mathematical tools like the t-test for analysis

Significance Level

Significance levels help us decide how much risk of being wrong we are willing to take in our hypothesis testing. In simpler terms, it tells us how confident we can be in the results. Commonly used significance levels are 1%, 5%, or 10%. For our study, the significance level is set at 1%, meaning there is a 1% chance that our results are due to random variation rather than the software patch making a true difference. This is a stringent level, ensuring that our conclusions are very robust.

• 1% significance level equates to higher confidence in the patch's effectiveness
• Helps in demonstrating whether observed effect is meaningful
• Balances the risk of making Type I errors (false positives)