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Chapter 3: Problem 9

If the average number of vacation days for a selection of various countries has a mean of 29.4 days and a standard deviation of 8.6 days, find the z scores for the average number of vacation days in each of these countries. \(\begin{array}{ll}{\text { Canada }} & {26 \text { days }} \\ {\text { Italy }} & {42 \text { days }} \\ {\text { United States } 13 \text { days }}\end{array}\)

Short Answer

Expert verified
Canada: \(-0.395\), Italy: \(1.465\), United States: \(-1.907\).

Step by step solution

01

Understanding Z-Score

The Z-score is a measure that describes a value's relation to the mean of a group of values. It is calculated by taking the difference between the value and the mean and then dividing by the standard deviation. The formula is given by:\[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
02

Calculate Z-Score for Canada

To find the Z-score for Canada:- The average vacation days are 26 days.- The mean, \(\mu\), is 29.4 days.- The standard deviation, \(\sigma\), is 8.6 days.Plug in the values into the formula:\[ Z = \frac{26 - 29.4}{8.6} = \frac{-3.4}{8.6} \approx -0.395 \]
03

Calculate Z-Score for Italy

To find the Z-score for Italy:- The average vacation days are 42 days.- The mean, \(\mu\), is 29.4 days.- The standard deviation, \(\sigma\), is 8.6 days.Plug in the values into the formula:\[ Z = \frac{42 - 29.4}{8.6} = \frac{12.6}{8.6} \approx 1.465 \]
04

Calculate Z-Score for the United States

To find the Z-score for the United States:- The average vacation days are 13 days.- The mean, \(\mu\), is 29.4 days.- The standard deviation, \(\sigma\), is 8.6 days.Plug in the values into the formula:\[ Z = \frac{13 - 29.4}{8.6} = \frac{-16.4}{8.6} \approx -1.907 \]

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Key Concepts

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

Mean
The term "mean" is commonly known as the average. It is a fundamental concept in statistics and mathematics. To find the mean, you add up all the numbers in a data set and then divide by the amount of numbers in that set. This gives you the central value of the data, representing a typical value.
The mean can provide a clear understanding of data by:
  • Indicating the central tendency of values.
  • Helping to compare different data sets.
  • Providing a basis for calculating other statistical measures, like the standard deviation and the Z-score.
The mean is sensitive to extremely high or low values, which can skew the result if the numbers are not evenly distributed. In real-world scenarios like calculating the average number of vacation days, the mean allows us to determine a baseline or expected value for comparison.
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are. It tells us how much the values deviate from the mean on average. A small standard deviation suggests that the values are close to the mean, while a large one indicates that the values are more dispersed.
Calculating the standard deviation involves:
  • Finding the mean of the set.
  • Subtracting the mean from each number to find the deviation.
  • Squaring these deviations to eliminate negative values.
  • Calculating the mean of these squared deviations.
  • Taking the square root of this mean.
In our vacation days example, the standard deviation helps us understand variance in vacation habits between different countries. A high standard deviation implies significant differences from the mean vacation days, indicating greater variability among the countries involved.
Vacation Days
Vacation days indicate how many days employees in a particular country can take off work within a year. This is an important measurement in comparing work-life balance and labor regulations across different cultures.
When we study vacation days through statistics, we can observe:
  • Work-related policies of various nations.
  • Differences in labor practices globally.
  • Impact of vacation time on lifestyle and productivity.
Here, the use of statistical measures like mean and standard deviation allows us to analyze and compare average vacation days in countries like Canada, Italy, and the United States, understanding how each compares to the international average.
Statistics
Statistics is the science of collecting, analyzing, interpreting, presenting, and organizing data. It is vital in drawing meaningful conclusions from data sets and supporting decision-making.
Key areas of statistics include:
  • Descriptive Statistics: Summarizes data points, usually in terms of age, height, or, as in this case, vacation days.
  • Inferential Statistics: Uses data from a sample to draw conclusions about a larger population.
By using statistics, we can gain insights into patterns and trends. In the context of vacation days, statistics allow us to assess and compare the average number of days people in different countries take off work per year, aiding in understanding international work-life balance.

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Most popular questions from this chapter

Which score indicates the highest relative position? a. A score of 3.2 on a test with \(\bar{X}=4.6\) and \(s=1.5\) b. A score of 630 on a test with \(\bar{X}=800\) and \(s=200\) c. A score of 43 on a test with \(\bar{X}=50\) and \(s=5\)

A measure to determine the skewness of a distribution is called the Pearson coefficient (PC) of skewness. The formula is $$\mathrm{PC}=\frac{3(\bar{X}-\mathrm{MD})}{s}$$ The values of the coefficient usually range from \(-3\) to \(+3 .\) When the distribution is symmetric, the coefficient is zero; when the distribution is positively skewed, it is positive; and when the distribution is negatively skewed, it is negative. Using the formula, find the coefficient of skewness for each distribution, and describe the shape of the distribution. a. Mean = 10, median = 8, standard deviation = 3. b. Mean = 42, median = 45, standard deviation = 4. c. Mean = 18.6, median = 18.6, standard deviation = 1.5. d. Mean = 98, median = 97.6, standard deviation = 4.

The average age of U.S. astronaut candidates in the past has been 34, but candidates have ranged in age from 26 to 46. Use the range rule of thumb to estimate the standard deviation of the applicants鈥� ages.

The data show the population (in thousands) for a recent year of a sample of cities in South Carolina. \(\begin{array}{llllll}{29} & {26} & {15} & {13} & {17} & {58} \\ {14} & {25} & {37} & {19} & {40} & {67} \\ {23} & {10} & {97} & {12} & {129} & {} \\\ {27} & {20} & {18} & {120} & {35} \\ {66} & {21} & {11} & {43} & {22}\end{array}\) Find the data value that corresponds to each percentile. a. 40th percentile b. 75th percentile c. 90th percentile d. 30th percentile Using the same data, find the percentile corresponding to the given data value. e. 27 f. 40 g. 58 h. 67

Describe which measure of central tendency鈥攎ean, median, or mode鈥攚as probably used in each situation. a. One-half of the factory workers make more than \(\$ 5.37\) per hour, and one- half make less than \(\$ 5.37\) per hour. b. The average number of children per family in the Plaza Heights Complex is 1.8 . c. Most people prefer red convertibles over any other color. d. The average person cuts the lawn once a week. e. The most common fear today is fear of speaking in public. f. The average age of college professors is 42.3 years.
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