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Chapter 2: Problem 59

If the largest observation is less than 1 standard deviation above the mean, then the distribution tends to be skewed to the left. If the smallest observation is less than 1 standard deviation below the mean, then the distribution tends to be skewed to the right. A professor examined the results of the first exam given in her statistics class. The scores were $$\begin{array}{llllllll} 35 & 59 & 70 & 73 & 75 & 81 & 84 & 86 \end{array}$$ The mean and standard deviation are 70.4 and 16.7 . Using these, determine whether the distribution is either left or right skewed. Construct a dot plot to check.

Short Answer

Expert verified
The distribution is skewed to the left.

Step by step solution

01

Calculate Largest Observation Deviation from Mean

The largest observation is 86. We need to determine how many standard deviations this is above the mean. Calculate \(86 - 70.4\) to find the deviation and divide by the standard deviation, 16.7: \[\frac{86 - 70.4}{16.7} = \frac{15.6}{16.7} \approx 0.93\]Since 0.93 is less than 1, the largest observation is less than 1 standard deviation above the mean.
02

Calculate Smallest Observation Deviation from Mean

The smallest observation is 35. Calculate \(70.4 - 35\) to find how many standard deviations it is below the mean, and divide by the standard deviation: \[\frac{70.4 - 35}{16.7} = \frac{35.4}{16.7}\approx 2.12\]Since 2.12 is greater than 1, the smallest observation is more than 1 standard deviation below the mean.
03

Interpret Skewness

According to the given criteria, if the largest observation is less than 1 standard deviation above the mean, the distribution tends to be skewed to the left. Since this condition is satisfied, we conclude that the distribution is skewed to the left.
04

Construct a Dot Plot (Optional Visual Check)

To construct a dot plot, plot each observation as a dot on a number line: ``` 30 40 50 60 70 80 90 . . . .. . . . ``` Observations: 35, 59, 70, 73, 75, 81, 84, 86. The dots show a clustering of data on the lower side, confirming a left skew.

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Key Concepts

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

Mean
The mean is a fundamental concept in statistics that represents the average of a set of numbers. To find the mean, we add up all the numbers and then divide by the quantity of those numbers. It's like finding the center or middle of your data.
For example, in our exercise, the scores were 35, 59, 70, 73, 75, 81, 84, and 86. To find the mean, we sum these up:
35 + 59 + 70 + 73 + 75 + 81 + 84 + 86 = 563.
Since there are 8 numbers, divide 563 by 8 to get the mean:
\( \frac{563}{8} = 70.4 \).
  • The mean helps us understand where the center of the data lies.
  • In our exercise, the mean is 70.4.

The positioning of the data around this mean tells us a lot about its distribution.
Standard Deviation
Standard deviation is like a measure of a party's vibe. It tells us how tightly packed the data is around the mean. A lower standard deviation means everyone is close together, while a higher one means people are spread out.
In simpler terms, it quantifies the amount of variation or dispersion in a dataset.
  • If data points spread out far from the mean, the standard deviation is high.
  • If data points are close to the mean, the standard deviation is low.

For the given data set: 35, 59, 70, 73, 75, 81, 84, and 86, the standard deviation is 16.7.
This tells us that, on average, the scores deviate from the mean by 16.7 points.
It's crucial for assessing the distribution, like seeing if the class did generally well or if there were wide differences.
Dot Plot
A dot plot is a simple way to visualize data. Each dot represents one occurrence of a value in your dataset. Dot plots are especially useful for small to medium-sized datasets when you want a quick, intuitive grasp of the distribution.
To create a dot plot, place a number line that spans the range of your data and put dots above the corresponding values.
For our data: 35, 59, 70, 73, 75, 81, 84, 86, you would plot it like this: ``` 30 40 50 60 70 80 90 . . . .. . . . ``` This arrangement visually shows how data points cluster or spread, indicating skewness.
  • Dot plots help in spotting outliers and understanding the data's spread.
  • They offer a straightforward visual confirmation of skewness.

In our case, we see more dots on the left, telling us it's skewed left.
Distribution Analysis
Distribution analysis helps us understand the overall shape and spread of data. It's like reading the personality of the dataset. When we analyze distribution, we look at aspects like symmetry, skewness, and kurtosis.
Skewness, for instance, tells us if the data is spread more to one side.
  • If most data points are on the low end, it's left-skewed (negatively skewed).
  • Right-skewed data has more points on the high end and a longer right tail.

In the exercise, by calculating the deviations from the mean using standard deviation, we determined the dataset's skewness.
A left skew means more data points lower than the mean.
Assessing distribution allows better predictions and decisions based on the data's overall trend.

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Most popular questions from this chapter

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?

According to the National Association of Home Builders, the median selling price of new homes in the United States in February 2014 was \(\$ 261,400\). Which of the following is the most plausible value for the standard deviation: \(-\$ 15,000, \$ 1000, \$ 60,000,\) or \(\$ 1,000,000 ?\) Why? Explain what's unrealistic about each of the other values.

The standard deviation, the range, and the interquartile range (IQR) summarize the variability of the data. a. Why is the standard deviation \(s\) usually preferred over the range? b. Why is the IQR sometimes preferred to \(s ?\) c. What is an advantage of \(s\) over the IQR?

Statistics published on www. allcountries.org based on figures supplied by the U.S. Census Bureau show that 24 fatal accidents or less were observed in \(23.1 \%\) of years from 1987 to 1999,25 or less in \(38.5 \%\) of years, 26 or less in \(46.2 \%\) of years, 27 or less in \(61.5 \%\) of years, 28 or less in \(69.2 \%\) of years, 29 or less in \(92.3 \%\) of years from 1987 to \(1999 .\) These are called cumulative percentages. a. What is the median number of fatal accidents observed in a year? Explain why. b. Nearly all the numbers of fatal accidents occurring from 1987 to 1999 fall between 17 and 37 . If the number of fatal accidents can be approximated by a bell-shaped curve, give a rough approximation for the standard deviation of the number of fatal accidents. Explain your reasoning.

During the 2010 Professional Golfers Association (PGA) season, 90 golfers earned at least $$\$ 1$$ million in tournament prize money. Of those, 5 earned at least $$\$ 4$$ million, 11 earned between $$\$ 3$$ million and $$\$ 4$$ million, 21 earned between $$\$ 2$$ million and $$\$ 3$$ million, and 53 earned between $$\$ 1$$ million and $$\$ 2$$ million. a. Would the data for all 90 golfers be symmetric, skewed to the left, or skewed to the right? b. Two measures of central tendency of the golfers' winnings were $$\$ 2,090,012$$ and $$\$ 1,646,853 .$$ Which do you think is the mean and which is the median?
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