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Chapter 10: Problem 47

Use the following information for the next five exercises. Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of 150 phones with OS1 had system failures within the first eight hours of operation. Nine out of another random sample of 150 phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1. What is the random variable?

Short Answer

Expert verified
The random variable is the count of system failures out of 150 phones for each OS.

Step by step solution

01

Define a Random Variable

A random variable is an outcome of a random process that can take on different values. In this context, we need to determine what specific event or measure we are examining as it relates to the system failures of the phone operating systems.
02

Identify the Outcome of Interest

We are comparing the proportions of system failures between two types of phone operating systems (OS1 and OS2). Thus, our outcome of interest is the proportion of system failures within the first eight hours for both operating systems.
03

Express the Random Variable Formally

The random variable, denoted by \( X \), represents the count of system failures in a group of phones. For OS1 and OS2, it counts the number of system failures out of the total sample of 150 phones.

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Key Concepts

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

Proportions
Proportions play a crucial role in statistics, especially when analyzing data collected from samples. In this context, we are interested in the proportion of system failures in two different types of operating systems. Proportion helps us understand the fraction of the total number of phones in each group that experienced system failures.
This can be mathematically expressed as the ratio of the number of observed system failures to the total number of phones tested. For OS1, the proportion is calculated by dividing the number of failures (15) by the total sample size (150). Similarly, for OS2, it's the number of failures (9) divided by the total samples (150).
  • Proportion for OS1: \( \frac{15}{150} \)
  • Proportion for OS2: \( \frac{9}{150} \)
By comparing these proportions, we can evaluate which operating system tends to have fewer system failures within the given timeframe.
System Failures
System failures, in this scenario, refer to crashes that occur within the first eight hours of using the phone's operating system. It is a crucial metric as it directly impacts user experience and reliability of the operating system. Crashes can occur due to various reasons like bugs, compatibility issues or overloads, but the critical part of our study is to quantify these failures.
The number of system failures is a countable event which can help in assessing the stability and efficiency of each operating system. In statistical terms, each system failure is considered a 'success' in the context of our study since we are counting this specific event over the trials (phones tested).
Understanding system failures helps manufacturers improve product quality and service. By assessing failure rates, developers can pinpoint weak spots, enhancing future software updates and designs.
Comparative Analysis
Comparative analysis involves evaluating two or more items to identify their similarities and differences. Here, our aim is to perform a comparative analysis of the system failure rates between OS1 and OS2. This is achieved by comparing the proportions of failures for each operating system.
By looking at the proportions from the previous section, we perform a statistical test (often a hypothesis test) to determine if the observed difference in proportions is significant or if it could have occurred by chance. This involves setting up null and alternative hypotheses. For instance, the null hypothesis might state that there is no difference in proportions of failures between the operating systems, while the alternative hypothesizes a difference.
  • Null Hypothesis: Proportion of OS1 failures = Proportion of OS2 failures
  • Alternative Hypothesis: Proportion of OS1 failures \( eq \) Proportion of OS2 failures
In conclusion, comparative analysis is essential in determining the better operating system in terms of stability and reliability, thus driving informed decisions or innovations.

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Most popular questions from this chapter

Use the following information to answer the next two exercises. An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a 鈥渂iofeedback exercise program.鈥� Six subjects were randomly selected and blood pressure measurements were recorded before and after the training. The difference between blood pressures was calculated (after - before) producing the following results: \(\overline{x}_{d}=-10.2 \mathrm{sd}=8.4\) Using the data, test the hypothesis that the blood pressure has decreased after the training. If \(\alpha=0.05,\) the \(p\) -value and the conclusion are a. 0.0014; There is sufficient evidence to conclude that the blood pressure decreased after the training. b. 0.0014; There is sufficient evidence to conclude that the blood pressure increased after the training. c. 0.0155; There is sufficient evidence to conclude that the blood pressure decreased after the training. d. 0.0155; There is sufficient evidence to conclude that the blood pressure increased after the training.

Use the following information to answer the next five exercises. A doctor wants to know if a blood pressure medication is effective. Six subjects have their blood pressures recorded. After twelve weeks on the medication, the same six subjects have their blood pressure recorded again. For this test, only systolic pressure is of concern. Test at the 1% significance level. $$ \begin{array}{|l|l|l|l|l|l|}\hline \text { Patient } & {\mathbf{A}} & {\mathbf{B}} & {\mathbf{C}} & {\mathbf{D}} & {\mathbf{E}} & {\mathbf{F}} \\\ \hline \text { Before } & {161} & {162} & {165} & {162} & {166} & {171} \\\ \hline \text { After } & {158} & {159} & {166} & {160} & {167} & {169} \\\ \hline\end{array} $$ State the null and alternative hypotheses.

Use the following information to answer the next 15 exercises: Indicate if the hypothesis test is for a. independent group means, population standard deviations, and/or variances known b. independent group means, population standard deviations, and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion The league mean batting average is 0.280 with a known standard deviation of 0.06. The Rattlers and the Vikings belong to the league. The mean batting average for a sample of eight Rattlers is 0.210, and the mean batting average for a sample of eight Vikings is 0.260. There are 24 players on the Rattlers and 19 players on the Vikings. Are the batting averages of the Rattlers and Vikings statistically different?

Use the following information to answer the next five exercises. A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller. $$\begin{array}{|l|l|l|}\hline \text { Plant Group } & {\text { Sample Mean Height of Plants (inches) }} & {\text { Population Standard Deviation }} \\\ \hline \text { Food } & {16} & {2.5} \\ \hline \text { No food } & {14} & {1.5} \\ \hline\end{array}$$ Is the population standard deviation known or unknown?

Use the following information to answer the next 15 exercises: Indicate if the hypothesis test is for a. independent group means, population standard deviations, and/or variances known b. independent group means, population standard deviations, and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion A new WiFi range booster is being offered to consumers. A researcher tests the native range of 12 different routers under the same conditions. The ranges are recorded. Then the researcher uses the new WiFi range booster and records the new ranges. Does the new WiFi range booster do a better job?
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