6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:38 minutesProblem 12.CR.6d
Textbook Question
Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).
d. If a vending machine is designed to accept quarters with weights above the 10th percentile P10 find the weight separating acceptable quarters from those that are not acceptable.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the weight that separates the bottom 10% of quarters (10th percentile, P10) from the rest, assuming the weights are normally distributed with a mean (μ) of 5.670 g and a standard deviation (σ) of 0.062 g.
Step 2: Recall the formula for a z-score in a normal distribution: z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation. We will use this formula to find the weight (X) corresponding to the 10th percentile.
Step 3: Use a z-table or statistical software to find the z-score corresponding to the 10th percentile. The z-score for the 10th percentile (P10) is approximately -1.28. This means that the value of X is 1.28 standard deviations below the mean.
Step 4: Rearrange the z-score formula to solve for X: X = μ + (z * σ). Substitute the known values: μ = 5.670 g, z = -1.28, and σ = 0.062 g.
Step 5: Perform the calculation to find X, which represents the weight separating acceptable quarters from those that are not acceptable. This value will be the 10th percentile weight.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the weights of quarters follow a normal distribution, which allows us to use statistical methods to determine percentiles.
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Percentiles
A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 10th percentile (P10) is the weight below which 10% of the quarter weights lie. Understanding percentiles is crucial for determining thresholds, such as the weight limit for acceptable quarters in the vending machine.
Z-scores
A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It allows for the comparison of scores from different distributions. To find the weight corresponding to the 10th percentile, one can calculate the Z-score for P10 and then use it to find the specific weight using the mean and standard deviation of the quarter weights.
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