The mean price of a pound of ground beef in 75 cities in the Midwest is $2.11 and the standard deviation is $0.56. A histogram of the data shows that the distribution is symmetric. A grocer is selling a pound of ground beef for $3.25. What is this price in standard units? Assuming the Empirical Rule applies, would this price be unusual or not? Round to the nearest hundredth.
A. $z=-2.04$; This price would not be unusual.
B. $z=2.04$; This is unusually expensive ground beef.
C. $z=2.04$; This price would not be unusual.
D. $z=-2.04$; This is unusually inexpensive ground beef.
Answer (2)
The z-score of the grocer's price of $3.25 is calculated to be approximately 2.04 , in d i c a t in g t ha tt hi s p r i ce i s u n u s u a ll ye x p e n s i v eco m p a re d t o t h e m e an p r i ce . A ccor d in g t o t h e E m p i r i c a lR u l e , a z − score g re a t er t han 2 i sco n s i d ere d u n u s u a l . T h u s , t h ecorrec t an s w er i s B : z=2.04$; This is unusually expensive ground beef.
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Calculate the z-score using the formula: z = σ x − μ .
Substitute the given values: x = 3.25 , μ = 2.11 , and σ = 0.56 .
Calculate the z-score: z = 0.56 3.25 − 2.11 ≈ 2.04 .
Since 2"> z = 2.04 > 2 , the price is unusually expensive: z = 2.04 .
Explanation
Understand the problem and provided data We are given the mean price of ground beef ( $2.11 ), the standard deviation ( $0.56 ), and a specific price ( $3.25 ). We need to find the z-score for this price and determine if it's unusual using the Empirical Rule.
Introduce the z-score formula The z-score formula is: z = σ x − μ where:
x is the observed value (the grocer's price),
μ is the mean,
σ is the standard deviation.
Calculate the z-score Now, we substitute the given values into the formula: z = 0.56 3.25 − 2.11 z = 0.56 1.14 z ≈ 2.0357
Round the z-score Rounding the z-score to the nearest hundredth, we get: z ≈ 2.04
Apply the Empirical Rule and conclude The Empirical Rule states that for a symmetric distribution, approximately 95% of the data falls within 2 standard deviations of the mean. Therefore, values with a z-score greater than 2 or less than -2 are considered unusual. Since our z-score is 2.04, which is greater than 2, this price is considered unusually expensive.
Examples
Understanding z-scores and the Empirical Rule can help you determine if a particular data point is unusual compared to the rest of the data. For example, in retail, if a store's sales are significantly higher than the average sales of similar stores (high z-score), it might indicate a successful marketing strategy or a unique product offering. Conversely, a very low z-score might signal problems that need attention. This kind of analysis is also used in finance to assess the risk and return of investments.
