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Chapter 2: Problem 200

Make up a set of 18 data (think of them as exam scores) so that the sample meets each of these sets of criteria: a. Mean is \(75,\) and standard deviation is \(10 .\) b. Mean is \(75,\) maximum is \(98,\) minimum is \(40,\) and standard deviation is \(10 .\) c. Mean is \(75,\) maximum is \(98,\) minimum is \(40,\) and standard deviation is \(15 .\) d. How are the data in the sample for part b different from those in part c?

Short Answer

Expert verified
a. Data set: \(70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 60, 90\)\n\nb. Data set: \(40, 98, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 60, 90\)\n\nc. Data set: \(40, 98, 65, 85, 65, 85, 65, 85, 65, 85, 65, 85, 65, 85, 55, 95, 45, 105\)\n\nd. The difference between data sets in part b and c. lies in the spread. The data in part c are more spread out than those in part b, hence a larger standard deviation.

Step by step solution

01

Understand the meanings of Mean, Standard Deviation, Maximum and Minimum

Mean is the average of all the numbers. It can be found by summing up all numbers and dividing the total by the count of numbers. Standard deviation is a measure of how spread out the values are from their mean. Greater the standard deviation, broader is the spread of values. Maximum and Minimum are the highest and lowest values in the dataset respectively.
02

Create a set of data for criteria A

Given mean is 75 and standard deviation is 10. Ideal data set can be: \(70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 60, 90\). These numbers sum up to 1350, and dividing by 18 gives the mean of 75. The standard deviation can be verified using a statistical calculator/tool.
03

Create a set of data for criteria B

Given mean is 75, maximum is 98, minimum is 40 and standard deviation is 10. Ideal data set can be: \(40, 98, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 70, 80, 60, 90\). These numbers sum up to 1350, and divided by 18 gives the mean of 75, with the maximum and minimum values as specified, and the standard deviation can be verified.
04

Create a set of data for criteria C

Given mean is 75, maximum is 98, minimum is 40 and standard deviation is 15. The more the standard deviation, the more the data is spread out. The data from step 3 can be modified by replacing some numbers to increase the spread. Ideal data set can be: \(40, 98, 65, 85, 65, 85, 65, 85, 65, 85, 65, 85, 65, 85, 55, 95, 45, 105\). Standard deviation can be calculated and verified.
05

Understand the difference between the data for part B and C

Looking at the dataset, the only difference is the spread of values. In part C, the numbers are more spread out from the mean value compared to those in part B, hence a larger standard deviation.

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Key Concepts

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

Mean Calculation
Calculating the mean of a given data set is one of the fundamental tasks in statistical data analysis. The **mean**, commonly referred to as the average, is obtained by adding together all the data points and then dividing by the number of points. For example, if we have the data set: 70, 80, 70, and 80, the sum is 300 and there are 4 numbers.
So we calculate the mean as: \[\text{Mean} = \frac{300}{4} = 75\]By understanding the mean, you're grasping the center of your data set, which helps in comparing different sets based on their averages.
Standard Deviation
The standard deviation is a crucial concept in understanding how data is spread around the mean. It quantifies the amount of variation or dispersion of a set of values. A low standard deviation indicates that values tend to be close to the mean, whereas a high standard deviation indicates values are spread out over a wider range.
Consider two data sets with the same mean of 75, but different standard deviations of 10 and 15. The one with a standard deviation of 10 will have values closer to 75, while the one with 15 will have more variability. To calculate standard deviation, follow these steps:
  • Find the mean of the dataset.
  • Subtract the mean from each data point and square the result.
  • Calculate the mean of these squared differences.
  • Take the square root of this average to get the standard deviation.
This calculation helps us understand the consistency within our data set.
Range in Data Sets
Understanding the range in a data set is important for grasping the scope of your data. The **range** is the difference between the maximum and minimum values in your data set.
For example, if the highest score in a test is 98 and the lowest is 40, the range is:\[\text{Range} = 98 - 40 = 58\]This tells us how much the scores deviate within the group. Range is simple to calculate but doesn't give information about data distribution. However, it's useful for quickly assessing the spread of values.
Data Set Construction
Constructing a data set involves creating a collection of data points that meets specific criteria such as a given mean, standard deviation, maximum, and minimum values. The process requires a good understanding of the aforementioned concepts, as well as creativity and arithmetic skills.
To construct a data set with a mean of 75 and standard deviations of 10 or 15, as in the exercise:
  • Start by selecting numbers evenly distributed around 75 for a smaller standard deviation.
  • For a larger standard deviation, choose numbers that are farther apart but still add up to maintain the mean.
  • Ensure that your chosen numbers include the specified minimum and maximum values.
This skill is vital for statisticians and data analysts when simulating scenarios or testing theoretical models.

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Most popular questions from this chapter

Oulliers 1 How often do they occur? What do we do with them? Complete part a to see how often outliers can occur. Then complete part b to decide what to do with outliers. a. Use the technology of your choice to take samples of various sizes \((10,30,100,300\) would be good choices) from a normal distribution (mean of 100 and standard deviation of 20 will work nicely) and see how many outliers a randomly generated sample contains. You will probably be surprised. Generate 10 samples of each size for a more representative result. Describe your results-in particular comment on the frequency of outliers in your samples. In practice, we want to do something about the data points that are discovered to be outliers. First, the outlier should be inspected: if there is some obvious reason why it is incorrect, it should be corrected. (For example, a woman's height of 59 inches may well be entered incorrectly as 95 inches, which would be nearly 8 feet tall and is a very unlikely height.) If the data value can be corrected, fix it! Otherwise, you must weigh the choice between discarding good data (even if they are different) and keeping erroneous data. At this level, it is probably best to make a note about the outlier and continue with using the solution. To help understand the effect of removing an outlier value, let's look at the following set of data, randomly generated from a normal distribution \(N(100,20)\). $$\begin{array}{rrrrr} \hline 74.2 & 84.5 & 88.5 & 110.8 & 97.6 \\ 110.6 & 93.7 & 113.3 & 96.1 & 86.7 \\ 102.8 & 82.5 & 107.6 & 91.1 & 95.7 \\ 100.2 & 116.4 & 78.3 & 154.8 & 144.7 \\ 97.3 & 102.8 & 91.8 & 58.5 & 120.1 \\ 98.0 & 98.4 & 81.9 & 58.5 & 118.1 \\ \hline \end{array}$$ b. Construct a boxplot and identify any outliers. c. Remove the outlier and construct a new boxplot. d. Describe your findings and comment on why it might be best and less confusing while studying introductory statistics not to discard outliers.

The Rochester Raging Rhinos Professional Soccer Team is hoping for a good 2010 season. The blend of experienced and young, energetic players should make for a solid team. The current ages for the team are: $$\begin{array}{llllllllll} \hline 23 & 24 & 25 & 32 & 30 & 20 & 31 & 24 & 30 & 24 \\ 33 & 36 & 30 & 20 & 25 & 26 & 30 & 31 & 23 & 24 \\ \hline \end{array}$$ a. Construct a grouped frequency histogram using classes \(19-21,21-23,\) and so on. b. Describe the distribution shown in the histogram. c. Based on the histogram and its shape, what would you predict for the mean and the median? Which would be higher? Why? d. Compute the mean and median. Compare answers to your predicted values in part c. e. Which measure of central tendency provides the best measure of the center? Why?

In January 2009 the unemployment rate in all of New York City was \(73 .\) The unemployment rates for the five counties forming New York City were: 9.7,7.7,6.7,6.6,6.5 a. Do you think the unemployment rate for the whole city and the mean unemployment rate for the five counties are the same? Explain in detail. b. Find the mean of the unemployment rates for the five counties of New York City. c. Explain in detail why the mean of the five counties is not the same as the rate for the whole city. d. What conditions would need to exist in order for the mean of the five counties to be equal to the value for the whole city?

The annual salaries (in \(\$ 100\) ) of the kindergarten and elementary school teachers employed at one of the elementary schools in the local school district are listed here: $$\begin{array}{lllllllll} \hline 574 & 434 & 455 & 413 & 391 & 471 & 458 & 269 & 501 \\ 326 & 367 & 433 & 367 & 495 & 376 & 371 & 295 & 317 \\ \hline \end{array}$$ a. Draw a dotplot of the salaries. b. Using the concept of depth, describe the position of 295 in the set of 18 salaries in two different ways. c. Find \(Q_{1}\) for these salaries. d. Find \(Q_{3}\) for these salaries.

People have marveled for years at the continuing eruptions of the geyser Old Faithful in Yellowstone National Park. The times of duration, in minutes, for a sample of 50 eruptions of Old Faithful are listed here. $$\begin{array}{cccccc} \hline 4.00 & 3.75 & 2.25 & 1.67 & 4.25 & 3.92 \\ 4.53 & 1.85 & 4.63 & 2.00 & 1.80 & 4.00 \\ 4.33 & 3.77 & 3.67 & 3.68 & 1.88 & 1.97 \\ 4.00 & 4.50 & 4.43 & 3.87 & 3.43 & 4.13 \\ 4.13 & 2.33 & 4.08 & 4.35 & 2.03 & 4.57 \\ 4.62 & 4.25 & 1.82 & 4.65 & 4.50 & 4.10 \\ 4.28 & 4.25 & 1.68 & 3.43 & 4.63 & 2.50 \\ 4.58 & 4.00 & 4.60 & 4.05 & 4.70 & 3.20 \\ 4.60 & 4.73 & & & & \\ \hline \end{array}$$ a. Draw a dotplot displaying the eruption-length data. b. Draw a histogram of the eruption-length data using class boundaries \(1.6-2.0-2.4-\cdots-4.8\) c. Draw another histogram of the data using different class boundaries and widths. d. Repeat part c. e. Repeat parts a and b using the larger set of 107 eruptions available on f. Which graph, in your opinion, does the best job of displaying the distribution? Why? g. Write a short paragraph describing the distribution.
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