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Chapter 2: Problem 80

Hamburger sales The manager of a fast-food restaurant records each day for a year the amount of money received from sales of food that day. Using software, he finds a bellshaped histogram with a mean of \(\$ 1165\) and a standard deviation of \(\$ 220 .\) Today the sales equaled \(\$ 2000\). Is this an unusually good day? Answer by providing statistical justification.

Short Answer

Expert verified
Yes, today is unusually good, as $2000 in sales is 3.80 standard deviations above the mean, which is rare for a normal distribution.

Step by step solution

01

Understanding the Problem

We need to determine if the sales of $2000 is unusually high compared to the usual sales. Given that the daily sales follow a normal distribution with a mean of $1165 and a standard deviation of $220, we can use Z-scores to find how many standard deviations $2000 is away from the mean.
02

Calculating the Z-Score

The Z-score formula is given by: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value in question (in this case, \(2000), \( \mu \) is the mean (\)1165), and \( \sigma \) is the standard deviation ($220). Substitute the values into the formula: \( Z = \frac{2000 - 1165}{220} \).
03

Performing the Calculation

Substitute and compute the Z-score: \( Z = \frac{2000 - 1165}{220} = \frac{835}{220} \approx 3.80 \).
04

Analyzing the Z-Score

A Z-score of approximately 3.80 indicates that $2000 is 3.80 standard deviations above the mean. In a normal distribution, a Z-score above 3 is considered rare and statistically significant, meaning this sales figure is unusually high because extreme values in a normal distribution typically have Z-scores greater than 3 or less than -3.

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Key Concepts

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

Z-score
A Z-score is a valuable statistical tool that allows us to standardize values within a dataset. It tells us how far and in what direction a value deviates from the mean of the dataset, measured in standard deviations.
This helps in assessing how unusual or typical a particular value is compared to the rest of the data.
  • The formula for calculating a Z-score is given by: \[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the specific data point, \( \mu \) is the mean of the data, and \( \sigma \) is the standard deviation.
  • A Z-score of 0 indicates that the data point is exactly at the mean.
  • If the Z-score is positive, it means the value is above the mean. Conversely, a negative Z-score signifies a value below the mean.
  • In our example, a Z-score of approximately 3.80 shows that today's sales are 3.80 standard deviations above the mean. This high Z-score suggests that sales of $2000 are not only above average but significantly higher than what's typically expected.
Standard Deviation
Standard deviation is a measure that tells us how spread out the values in a dataset are around the mean. A low standard deviation means that most of the numbers are very close to the average, while a high standard deviation indicates that the numbers are spread out over a wider range.
  • Formulaically, the standard deviation \( \sigma \) is derived as the square root of the variance.
  • In our given scenario, the standard deviation of \(220 tells us that most of the daily sales figures tend to cluster around the mean of \)1165.
  • This clustering implies that when something outside this range occurs, such as a Z-score of 3.80, it stands out as exceptionally high compared to the usual variability.
  • Understanding the standard deviation is essential, as it helps in calculating the Z-score, giving quantitative insights into how concentrated the data points are.
Bell-Shaped Histogram
A bell-shaped histogram is a visual representation of data that forms a bell curve, which is symmetrical and centered around the mean. This is characteristic of a normal distribution, where most of the data points fall near the mean, with fewer points appearing as extremes.
  • The shape of this histogram is useful because it provides a clear picture of how data is distributed across a range of values.
  • In a perfectly normal distribution, the mean, median, and mode all line up, falling right at the center of the bell curve.
  • The value of the bell-shaped histogram is that it allows us to easily spot standard deviations, which helps in identifying how extreme certain data points are.
  • With the sales data forming a bell-shaped histogram, we can infer that the $2000 sale is an anomaly sitting at the higher extreme of the sales distribution range.

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Most popular questions from this chapter

Number of children For the question "How many children have you ever had?" in the 2008 General Social Survey, the results were $$ \begin{array}{lccccccccc} \text { No. children } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8+ \\ \text { Count } & 521 & 323 & 524 & 344 & 160 & 77 & 30 & 19 & 22 \end{array} $$ a. Which is the most appropriate graph to display the data-dot plot, stem-and- leaf plot, or histogram? Why? b. Based on sketching or using software to construct the graph, characterize this distribution as skewed to the left, skewed to the right, or symmetric. Explain.

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How many friends? \(\quad\) A recent General Social Survey asked respondents how many close friends they had. For a sample of 1467 people, the mean was 7.4 and the standard deviation was 11.0 . The distribution had a median of 5 and a mode of 4. (Source: Data from CSM, UCBerkeley.) a. Based on these statistics, what would you surmise about the shape of the distribution? Why? b. Does the empirical rule apply to these data? Why or why not?

Teachers' salaries According to Statistical Abstract of the United States, \(2006,\) average salary (in dollars) of secondary school classroom teachers in 2004 in the United States varied among states with a five-number summary of: minimum \(=33,100, \mathrm{Q} 1=39,250,\) Median \(=42,700\), \(\mathrm{Q} 3=48,850,\) Maximum \(=61,800 .\) a. Find and interpret the range and interquartile range. b. Sketch a box plot, marking the five-number summary on it. c. Predict the direction of skew for this distribution. Explain. d. If the distribution, although skewed, is approximately bell shaped, which of the following would be the most realistic value for the standard deviation: (i) 100 (ii) 1000 , (iii) \(6000,\) or (iv) \(25,000 ?\) Explain your reasoning.

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