**Educational Website Transcription** ### Exercises and Concepts 1. **Find the Probability** - **e.** Find \( P(x < 6) \). - **f.** Find \( P(6.5 \leq x \leq 7.25) \). - **g.** What is the probability that each of the next six bottles filled by the new machine will contain more than 7.25 ounces of beverage? Assume that the amount of beverage dispensed in one bottle is independent of the amount dispensed in another bottle. 2. **Time Delays at a Bus Stop** A bus is scheduled to stop at a certain bus stop every half hour, on the hour and the half hour. At the end of the day, buses still stop every 30 minutes, but due to frequent earlier delays, the bus is never early and is likely to be late. The director of the bus line claims that the length of time a bus is late is uniformly distributed and the maximum time that a bus is late is 20 minutes. - **a.** If the director's claim is true, what is the expected number of minutes a bus will be late? - **b.** If the director's claim is true, what is the probability that the last bus on a given day will be more than 19 minutes late? - **c.** If you arrive at the bus stop at the end of a day at exactly half-past the hour and must wait more than 19 minutes for the bus, what would you conclude about the director’s claim? Why? 3. **Applying the Concepts—Advanced** **Gouges on a Spindle** A tool-and-die machine shop produces extremely high-tolerance spindles. The spindles are 18-inch slender rods used in a variety of military equipment. A piece of equipment used in the manufacture of... (Text continues in exercise 5.19.) ### Explanation of Concepts - **Uniform Distribution:** This is a probability distribution where all outcomes are equally likely within a certain range. For the bus delay scenario, the delay is uniformly distributed between 0 to 20 minutes. - **Expected Value:** This is the average outcome if an experiment is repeated many times. For a uniform distribution over a range [a, b], the expected value is \((a + b) / 2\). These exercises involve applying concepts

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Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related. 

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.