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Chapter 3: Problem 9

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, \((b)\) median, (c) mode, (d) midrange, and then answer the given question. Listed below are the numbers of Atlantic hurricanes that occurred in each year. The data are listed in order by year, starting with the year 2000 . What important feature of the data is not revealed by any of the measures of center? $$ \begin{array}{llllllllllllll} 8 & 9 & 8 & 7 & 9 & 15 & 5 & 6 & 8 & 4 & 12 & 7 & 8 & 2 \end{array} $$

Short Answer

Expert verified
Mean = 7.71, Median = 8, Mode = 8, Midrange = 8.5. Outliers are not revealed.

Step by step solution

01

- Find the Mean

To find the mean, sum all the numbers and then divide by the number of values.The numbers are: 8, 9, 8, 7, 9, 15, 5, 6, 8, 4, 12, 7, 8, and 2.Sum: 8 + 9 + 8 + 7 + 9 + 15 + 5 + 6 + 8 + 4 + 12 + 7 + 8 + 2 = 108There are 14 numbers.Mean = Sum / Number of values = 108 / 14 = 7.71 (rounded to two decimal places)
02

- Find the Median

To find the median, first sort the numbers in ascending order and then find the middle number. If there are an even number of observations, the median is the average of the two middle numbers.Sorted numbers: 2, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 9, 12, 15.Since there are 14 numbers, the median is the average of the 7th and 8th numbers.Median = (8 + 8) / 2 = 8
03

- Find the Mode

The mode is the number that appears most frequently.In the sorted list, 2, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 9, 12, 15, the number 8 appears the most (four times).Mode = 8
04

- Find the Midrange

The midrange is the average of the maximum and minimum numbers in the list.Maximum number = 15Minimum number = 2Midrange = (Maximum + Minimum) / 2 = (15 + 2) / 2 = 17 / 2 = 8.5
05

- Identify the Important Feature Not Revealed

The measures of center (mean, median, mode, midrange) do not reveal the presence of any potential outliers in the data set. In this case, the number 15 could be considered an outlier, as it is significantly higher than the other values.

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Key Concepts

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

Mean
In statistics, the mean is a measure of central tendency that represents the average value of a data set. Here鈥檚 how you calculate it:
1. Add up all the numbers. For our hurricane example: \(8 + 9 + 8 + 7 + 9 + 15 + 5 + 6 + 8 + 4 + 12 + 7 + 8 + 2 = 108\)
2. Divide the sum by the number of values. There are 14 values, so the mean is \( \frac{108}{14} = 7.71\)
Mean: 7.71
Median
The median is a measure of central tendency that represents the middle value of a dataset. To find it:
1. Sort the numbers in ascending order. For the hurricane example: 2, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 9, 12, 15
2. If the number of values is odd, the median is the middle number. If even, it鈥檚 the average of the two middle numbers. Here, the middle numbers are the 7th and 8th values (both 8). So, \( \frac{8 + 8}{2} = 8 \)
Median: 8
Mode
The mode is the value that appears most frequently in a dataset. For our hurricane example, sorting the list gives: 2, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 9, 12, 15. The number 8 appears the most (four times), so the mode is:
Mode: 8
Midrange
The midrange is a measure of central tendency that calculates the average of the maximum and minimum values in a dataset. For our hurricane numbers:
1. Identify the maximum value (15) and the minimum value (2).
2. Calculate the average of these two values: \( \frac{15 + 2}{2} = 8.5 \)
Midrange: 8.5
Statistical Outliers
Outliers are values that are significantly higher or lower than the other values in a dataset. They can affect various measures of central tendency, especially the mean. In our hurricane dataset, the number 15 could be an outlier as it is much higher than the other numbers. Measures like the mean, median, mode, and midrange might not reveal these outliers. To identify them, more advanced statistical methods or visual tools like box plots can be helpful.

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Most popular questions from this chapter

Refer to the frequency distribution in the given exercise and find the standard deviation by using the formula below, where \(x\) represents the class midpoint, \(f\) represents the class frequency, and \(n\) represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 11.5 years; (Exercise 38) 8.9 years; (Exercise 39) 59.5; (Exercise 40) 65.4. $$s=\sqrt{\frac{n\left[\Sigma\left(f \cdot x^{2}\right)\right]-[\Sigma(f \cdot x)]^{2}}{n(n-1)}}$$ $$ \begin{array}{c|c} \hline \begin{array}{c} \text { Blood Platelet } \\ \text { Count of } \\ \text { Females } \end{array} & \text { Frequency } \\ \hline 100-199 & 25 \\ \hline 200-299 & 92 \\ \hline 300-399 & 28 \\ \hline 400-499 & 0 \\ \hline 500-599 & 2 \\ \hline \end{array} $$

A student of the author earned grades of \(63,91,88,84\), and 79 on her five regular statistics tests. She earned grades of 86 on the final exam and 90 on her class projects. Her combined homework grade was \(70 .\) The five regular tests count for \(60 \%\) of the final grade, the final exam counts for \(10 \%\), the project counts for \(15 \%\), and homework counts for \(15 \%\). What is her weighted mean grade? What letter grade did she earn (A, B, C, D, or F)? Assume that a mean of 90 or above is an \(\mathrm{A}\), a mean of 80 to 89 is a \(\mathrm{B}\), and so on.

The 20 brain volumes \(\left(\mathrm{cm}^{3}\right)\) from Data Set 8 "IQ and Brain Size" in Appendix B have a mean of \(1126.0 \mathrm{~cm}^{3}\) and a standard deviation of \(124.9 \mathrm{~cm}^{3}\). Use the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of \(1440 \mathrm{~cm}^{3}\) be significantly high?

Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.) a. Find the variance \(\sigma^{2}\) of the population \(\\{9\) cigarettes, 10 cigarettes, 20 cigarettes \(\\}\). b. After listing the nine different possible samples of two values selected with replacement, find the sample variance \(s^{2}\) (which includes division by \(n-1\) ) for each of them; then find the mean of the nine sample variances \(s^{2}\). c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by \(n\) ); then find the mean of those nine population variances. d. Which approach results in values that are better estimates of \(\sigma^{2}\) : part (b) or part (c)? Why? When computing variances of samples, should you use division by \(n\) or \(n-1\) ? e. The preceding parts show that \(s^{2}\) is an unbiased estimator of \(\sigma^{2}\). Is \(s\) an unbiased estimator of \(\sigma\) ? Explain.

Find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as "minutes") in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions. Listed below are the weights in pounds of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII (the same players from the preceding exercise). Are the measures of variation likely to be typical of all NFL players? $$ \begin{array}{lllllllllll} 189 & 254 & 235 & 225 & 190 & 305 & 195 & 202 & 190 & 252 & 305 \end{array} $$
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