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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 2 - Descriptive StatisticsProblem 2.1.45c
Chapter 2, Problem 2.1.45c

What Would You Do? You work at a bank and are asked to recommend the amount of cash to put in an ATM each day. You do not want to put in too much (which would cause security concerns) or too little (which may create customer irritation). The daily withdrawals (in hundreds of dollars) for 30 days are listed. 72 84 61 76 104 76 86 92 80 88 98 76 97 82 84 67 70 81 82 89 74 73 86 81 85 78 82 80 91 83
If you are willing to run out of cash on 10% of the days, how much cash should you put in the ATM each day? Explain.

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Step 1: Organize the data. The daily withdrawals are given as a list of 30 values. Arrange these values in ascending order to make it easier to calculate percentiles.
Step 2: Determine the 90th percentile. Since you are willing to run out of cash on 10% of the days, you need to calculate the 90th percentile of the data. The 90th percentile is the value below which 90% of the data falls.
Step 3: Use the formula for the percentile position. The position of the 90th percentile in the ordered data can be calculated using the formula: P = (n + 1) * (percentile / 100), where n is the number of data points (30 in this case) and the percentile is 90.
Step 4: Locate the value corresponding to the calculated position. If the position is an integer, the value at that position in the ordered data is the 90th percentile. If the position is not an integer, interpolate between the two closest values in the ordered data.
Step 5: Use the 90th percentile value as the recommended amount of cash to put in the ATM each day. This ensures that the ATM will have enough cash for 90% of the days, with a 10% chance of running out.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Descriptive Statistics

Descriptive statistics summarize and describe the main features of a dataset. In this context, measures such as the mean, median, and standard deviation of daily withdrawals can provide insights into typical cash demands and variability. Understanding these statistics helps in making informed decisions about how much cash to stock in the ATM.
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Parameters vs. Statistics

Percentiles

Percentiles are used to understand the distribution of data by indicating the value below which a given percentage of observations fall. For example, if you want to ensure that the ATM runs out of cash only 10% of the time, you would look for the 90th percentile of the daily withdrawals. This value represents the cash amount that should be stocked to meet the demand on most days.

Risk Management

Risk management involves identifying, assessing, and prioritizing risks followed by coordinated efforts to minimize, monitor, and control the probability of unfortunate events. In this scenario, balancing the risk of running out of cash (customer dissatisfaction) against security concerns of having too much cash is crucial. Effective risk management strategies will help determine the optimal cash amount to stock in the ATM.
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Step 4: State Conclusion
Related Practice
Textbook Question

Using and Interpreting Concepts


Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12,

(c) identify any outliers.


56 63 51 60 57 60 60 54 63 59 80 63 60 62 65

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Textbook Question

Use the data set and the indicated number of classes to construct

(c) a frequency polygon,


Hospitals

Number of classes: 8

Data set: Number of hospitals in each of the 50 U.S. states and 5 inhabited territories (Source: American Hospital Directory) 10 90 51 1 77 341 56 34 8 214 111 3 14 40 18 142 102 55 75 108 72 53 19 105 55 83 1 69 19 108 10 27 14 78 37 31 186 146 90 37 177 52 11 67 25 100 361 35 91 2 7 61 78 33 14

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Textbook Question

Use the ogive to approximate the

the number of black bears that weigh between 158.5 pounds and 244.5 pounds.

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Textbook Question

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.


(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.


i.

ii.

iii.

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Textbook Question

Studying Refer to the data set in Exercise 23 and the box-and-whisker plot you drew that represents the data set.


c. You randomly select one student from the sample. What is the likelihood that the student studied less than 2 hours per day? Write your answer as a percent.

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Textbook Question

Pearson’s Index of Skewness The English statistician Karl Pearson (1857–1936) introduced a formula for the skewness of a distribution.

P = 3 (x̄ - median) / s

Most distributions have an index of skewness between -3 and 3. When P > 0, the data are skewed right. When P < 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.


c. x̄ = 9.2, s = 1.8, median = 9.2

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