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Chapter 2: Problem 33

In New Zealand, the mean and median weekly earnings for males in 2009 was \(\$ 993\) and \(\$ 870\), respectively and for females, the mean and median weekly earnings were \(\$ 683\) and \(\$ 625\), respectively (www.nzdotstat.stats.govt.nz). Does this suggest that the distribution of weekly earnings for males is symmetric, skewed to the right, or skewed to the left? What about the distribution of weekly earnings for females? Explain.

Short Answer

Expert verified
Both distributions are skewed to the right.

Step by step solution

01

Understanding the Problem

We need to determine the skewness of the distributions of weekly earnings for males and females. This involves comparing the mean and median values.
02

Reviewing Key Concepts

In statistics, if the mean is greater than the median, the distribution is typically skewed to the right (positively skewed). If the mean is less than the median, the distribution is skewed to the left (negatively skewed). If the mean is approximately equal to the median, the distribution is likely symmetric.
03

Analyzing Male Earnings

For males, the mean weekly earnings is \(\\(993\) and the median is \(\\)870\). Since the mean is greater than the median, it suggests that the distribution is skewed to the right.
04

Analyzing Female Earnings

For females, the mean weekly earnings is \(\\(683\) and the median is \(\\)625\). Similarly, because the mean is greater than the median, the distribution is also skewed to the right.
05

Conclusion

Both distributions, for males and females, are skewed to the right, indicating that a larger number of individuals earn less than the average, with a tail extending to higher earnings.

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Key Concepts

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

Mean and Median Comparison
Understanding the comparison between mean and median is essential to interpret statistical data.
The mean is the arithmetic average of a set of numbers. It is calculated by summing all the data points and dividing by the number of data points. On the other hand, the median is the middle value when data points are arranged in order. Thus, the median separates the lower half from the upper half of the dataset.

One key aspect of comparing mean and median is determining the shape of the data distribution.
  • If the mean is greater than the median, the data is likely skewed to the right.
  • If the mean is less than the median, it tends to be skewed to the left.
  • If the mean and median are close to each other, it suggests a symmetric distribution.
Thus, inspecting the mean and median can provide valuable insights into the distribution's nature and any potential outliers affecting the data.
Skewness in Data
Skewness is a measure of asymmetry in a data distribution.

A distribution can be categorized based on the direction and degree of its skewness:
  • Right skewed (positively skewed) distributions have a longer tail on the right. This occurs when the mean is greater than the median, often indicating that high outliers are stretching the distribution.
  • Left skewed (negatively skewed) distributions have a longer tail on the left. Here, the mean is typically less than the median, due to the presence of low outliers.
When a distribution is skewed, it implies that the majority of data points are concentrated on one side of the scale, with a tail extending in the opposite direction. Recognizing skewness is crucial for making decisions based on data insights, as it affects measures like the mean, which might not accurately represent the center of the data.
Distribution Symmetry
Symmetrical distributions are those where the left and right sides of the histogram are mirror images of each other.

In a perfectly symmetric distribution, the mean, median, and mode all share the same value, marking the center point of the data. This commonly occurs in distributions like the normal (Gaussian) distribution.

Properties of a symmetric distribution:
  • The data is evenly spread around the mean.
  • Outliers, if present, are balanced on both sides of the center.
  • No skewness: the tails on either side of the mean are of equal length.
Symmetric distributions are often simpler to analyze because measures of central tendency are consistent with one another. Recognizing this symmetry can simplify statistical inference, as many statistical tests assume normality or symmetry in the data.

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Most popular questions from this chapter

For each of the following variables, indicate whether you would expect its histogram to be symmetric, skewed to the right, or skewed to the left. Explain why. a. The price of a certain model of smartwatch in different stores in your district b. The amount of time students use to take an exam in your school c. The grade point average (GPA) in your academic program this year d. The salary of all the employees in a company.

A student asked her coworkers, parents, and friends, "How many friends do you have on Facebook?" She summarized her data and reported that the average number of Facebook friends in her sample is 170 with a standard deviation of \(90 .\) The distribution had a median of 120 and a mode of 105 . 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?

Bad statistic \(\quad\) A teacher summarizes grades on an exam by \(\operatorname{Min}=26, \mathrm{Q} 1=67, \mathrm{Q} 2=80, \mathrm{Q} 3=87, \operatorname{Max}=100\) Mean \(=76,\) Mode \(=100,\) Standard deviation \(=76\) \(\mathrm{IQR}=20\) She incorrectly recorded one of these. Which one do you think it was? Why?

A study of 13 children suffering from asthma (Clinical and Experimental Allergy, vol. \(20,\) pp. \(429-432,1990\) ) compared single inhaled doses of formoterol (F) and salbutamol (S). Each child was evaluated using both medications. The outcome measured was the child's peak expiratory flow (PEF) eight hours following treament. Is there a difference in the PEF level for the two medications? The data on PEF follow: $$ \begin{array}{ccc} \hline \text { Child } & \mathbf{F} & \mathbf{S} \\ \hline 1 & 310 & 270 \\ 2 & 385 & 370 \\ 3 & 400 & 310 \\ 4 & 310 & 260 \\ 5 & 410 & 380 \\ 6 & 370 & 300 \\ 7 & 410 & 390 \\ 8 & 320 & 290 \\ 9 & 330 & 365 \\ 10 & 250 & 210 \\ 11 & 380 & 350 \\ 12 & 340 & 260 \\ 13 & 220 & 90 \\ \hline \end{array} $$ a. Construct plots to compare formoterol and salbutamol. Write a short summary comparing the two distributions of the peak expiratory flow. b. Consider the distribution of differences between the PEF levels of the two medications. Find the 13 differences and construct and interpret a plot of the differences. If on the average there is no difference between the PEF level for the two brands, where would you expect the differences to be centered?

Classify each of the following variables as categorical or quantitative. a. The number of social media accounts you have (Facebook, Twitter, LinkedIn, Instagram, etc.) b. Preferred soccer team c. Choice of smartphone model to buy d. Distance (in kilometers) of commute to work
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