Textbook Question
Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.
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6. Normal Distribution and Continuous Random Variables
Probabilities & Z-Scores w/ Graphing Calculator
1:50 minutesProblem 2.R.46
Textbook Question
The towing capacities (in pounds) of all the pickup trucks at a dealership have a bell-shaped distribution, with a mean of 11,830 pounds and a standard deviation of 2370 pounds. In Exercises 45– 48, use the corresponding z-score to determine whether the towing capacity is unusual. Explain your reasoning.
5,500 pounds
Verified step by step guidance1
Step 1: Understand the problem. The towing capacities follow a bell-shaped distribution, which indicates a normal distribution. The mean (μ) is 11,830 pounds, and the standard deviation (σ) is 2,370 pounds. We are tasked with determining if a towing capacity of 5,500 pounds is unusual using the z-score formula.
Step 2: Recall the z-score formula: , where x is the data point, μ is the mean, and σ is the standard deviation. The z-score measures how many standard deviations a data point is from the mean.
Step 3: Substitute the given values into the formula. Here, x = 5,500, μ = 11,830, and σ = 2,370. The z-score calculation will look like this: .
Step 4: Interpret the z-score. A z-score less than -2 or greater than 2 is typically considered unusual in a normal distribution. This is because approximately 95% of the data falls within 2 standard deviations of the mean, leaving only 5% of the data in the tails.
Step 5: After calculating the z-score, compare its value to the threshold of -2 or 2. If the z-score is outside this range, the towing capacity of 5,500 pounds is considered unusual. If it is within this range, it is not unusual.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
A normal distribution is a bell-shaped probability distribution characterized by its mean and standard deviation. In this context, the towing capacities of pickup trucks follow a normal distribution, meaning most values cluster around the mean (11,830 pounds) and taper off symmetrically towards the extremes. Understanding this concept is crucial for determining how unusual a specific towing capacity is relative to the overall distribution.
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Z-Score
The z-score is a statistical measure that indicates how many standard deviations an element is from the mean of a distribution. It is calculated by subtracting the mean from the value in question and then dividing by the standard deviation. In this case, calculating the z-score for a towing capacity of 5,500 pounds will help assess whether this value is considered unusual within the context of the truck towing capacities.
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Unusual Values
In statistics, a value is often considered unusual if it lies more than two standard deviations away from the mean in a normal distribution. This threshold helps identify outliers or extreme values that may not fit the general pattern of the data. By determining the z-score for the towing capacity of 5,500 pounds, we can evaluate whether it falls within this unusual range, thus informing our reasoning about its significance.
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