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One of the most profitable items at Al's Auto Security Shop is the remote starting system. Let \(x\) be the number of such systems installed on a given day at this shop. The following table lists the frequency distribution of \(x\) for the past 80 days. $$ \begin{array}{l|ccccc} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f & 8 & 20 & 24 & 16 & 12 \\ \hline \end{array} $$ a. Construct a probability distribution table for the number of remote starting systems installed on a given day. b. Are the probabilities listed in the table of part a exact or approximate probabilities of various outcomes? Explain. c. Find the following probabilities. i. \(P(3)\) ii. \(P(x \geq 3)\) iii. \(P(2 \leq x \leq 4)\) iv. \(P(x<4)\)

Step by step solution

01

Construct the Probability Distribution

Firstly, find the sum of all frequencies given in the table. Find the probability of each event by dividing its frequency by the total. This is done by calculating \(P(x) = \frac{f}{80}\) for each value of \(x\). Create a new table with these values.

02

Determine Type of Probabilities

The probabilities are exact because they are based on the past data from 80 days. They represent the exact distribution of systems installed on a specific number of days.

03

Calculate Individual Probabilities

Calculate each probability with the following formulas: \(P(3) = P(x=3)\), \(P(x \geq 3) = P(x=3) + P(x=4) + P(x=5)\), \(P(2 \leq x \leq 4) = P(x=2) + P(x=3) + P(x=4)\), \(P(x<4) = P(x=1) + P(x=2) + P(x=3)\). Refer to the probability distribution table created in step 1 for calculating these probabilities.

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Key Concepts

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

Frequency Distribution

Understanding frequency distribution is a key aspect when you want to analyze data sets. In the context of Al's Auto Security Shop, the frequency distribution tells us how often each number of remote start systems was installed over 80 days. Each frequency corresponds to a specific number of installations.
This is represented in the table with frequencies for zero up to five installations per day. The sum of these frequencies should equal the total number of observed instances, which is 80 days in this case.

• Identifies how data points are distributed across different outcomes.
• Uses frequencies to show which results are more common or less common.
• Helpful in constructing probability distributions, like determining the probability of each event occurring.

Calculating and analyzing frequency distribution helps in understanding patterns or trends in data effectively.

Calculating Probabilities

To calculate probabilities from a frequency distribution, you use the frequency of each event divided by the total number of observations. This gives you the probability of each event. For Al's Auto Security Shop, if a system was installed 24 times when 3 systems were installed on a given day, and there were 80 days in total, then the probability for three installations in one day is \(P(3) = \frac{24}{80}\).
Understanding the importance of calculating these probabilities involves:

• Converting frequencies into a ratio of occurrence over the total.
• Ensuring each probability is between 0 and 1, with the sum equal to 1.
• Allowing users to predict future outcomes based on past data.

Using these probabilities, different questions can be answered about the likelihood of future events, offering insight into decision-making and strategy for potential outcomes.

Exact and Approximate Probabilities

Exact probabilities are determined using known, definitive data. In the case of Al's Auto Shop, the probabilities are exact, as they are derived from actual observed data over 80 days. This certainty means that the calculated probabilities represent real past outcomes.
Exact probabilities have certain critical features:

• They provide a true measure of likelihood based on historical data.
• Offer reliable insights as they reflect actual occurrences rather than estimations.
• Are advantageous in decision-making processes because they reduce assumptions.

Exact probabilities are factual, increasing the reliability of predictions and analysis compared to approximate probabilities, which are often estimated using sample data and can include a margin of error.

Probability Formulas

Understanding the use of probability formulas is crucial when analyzing data. For Al's Auto Security Shop, these formulas help calculate events' probabilities based on frequency distributions. Formulas such as:

• \[P(3) = \frac{f(3)}{N} = \frac{24}{80}\]
• \[P(x \geq 3) = P(3) + P(4) + P(5)\]
• \[P(2 \leq x \leq 4) = P(2) + P(3) + P(4)\]
• \[P(x < 4) = P(1) + P(2) + P(3)\]

These are applied to calculate specific probabilities such as cumulative or range probabilities.
Using these formulas allows comprehensive analysis for different scenarios, contributing significantly to predictive modeling and effective planning, especially with real-world applications like sales projections or inventory management.