/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 3.48 The accompanying data on pe... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 48

3.48 The accompanying data on percentage of juice lost after thawing for 19 different strawberry varieties appeared in the article "Evaluation of Strawberry Cultivars with Different Degrees of Resistance to Red Scale" (Fruit Varieties Journal [1991]: \(12-17\) ). $$ \begin{array}{lllllllllll} 46 & 51 & 44 & 50 & 33 & 46 & 60 & 41 & 55 & 46 & 53 \\ 53 & 42 & 44 & 50 & 54 & 46 & 41 & 48 & & & \end{array} $$ a. Are there any outliers? If so, which values are outliers? b. Construct a boxplot, and comment on the important features of the plot.

Short Answer

Expert verified
There are no outliers in this dataset, the values all fall within accepted range. The boxplot of the data indicates a slightly right-skewed distribution, with the median value being 48.

Step by step solution

01

Calculate the quartiles

Order the data in increasing number and divide it into quarters. The first quartile (Q1) is the value between the smallest number and the median, and the third quartile (Q3) is the value between the median and the highest number. In this exercise, after ordering the data: \(33, 41, 41, 42, 44, 44, 46, 46, 46, 48, 50, 50, 51, 53, 53, 54, 55, 60\), Q1 (25th percentile) is \(44\) and Q3 (75th percentile) is \(53\).
02

Find the interquartile range (IQR) and detect Outliers

IQR is calculated by subtracting Q1 from Q3. IQR for this dataset will be \(53-44 = 9\). Any data points, which are smaller than \(Q1 - 1.5*IQR\) or larger than \(Q3 + 1.5*IQR\) are considered as outliers. Hence the outliers range becomes \(44 - 1.5*9 = 30.5\) to \(53 + 1.5*9 = 66.5\) . Hence there are no outliers as the minimum value is \(33\) and the maximum value is \(60\) which lies in this range.
03

Construct a boxplot

A boxplot graphically represents the distribution of data based on a five-number summary (minimum, first quartile Q1, median, third quartile Q3, and maximum). For the given dataset, graph an axis from the minimum value (33) to the maximum value (60), then draw a box from the first quartile (44) to the third quartile (53) and draw a line at the median (48).
04

Analyze the boxplot

The boxplot's box represents values from Q1 to Q3, the line inside the box is the median. The 'whiskers' (lines protruding from the box) shows the spread of the rest of the data. A symmetrical boxplotindicates normal distribution. In this case, the plot may show a little skewness to the right as the median line is closer to quartile 1, however, not significant enough to conclude on skewness or asymmetry. There are no outliers in this boxplot.

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Key Concepts

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

Quartiles
Quartiles are a fundamental concept in statistics, used to divide a dataset into four equal parts. Here鈥檚 how:
  • First Quartile (Q1): Known also as the 25th percentile. It marks the value below which 25% of the data falls.
  • Second Quartile (Median): The 50th percentile, the median is the middle value of a dataset.
  • Third Quartile (Q3): The 75th percentile, where 75% of the data is below.
To find Q1 and Q3, order the data from smallest to largest. For our strawberry dataset, after arranging the values, Q1 is 44, and Q3 is 53. Quartiles help understand the distribution and spread of the data, revealing more about its central tendency and variability.
Interquartile Range
The Interquartile Range (IQR) is the measure of variability that lies between the first quartile (Q1) and the third quartile (Q3). It essentially captures the middle 50% of the data.
  • Formula: IQR = Q3 - Q1
  • For our dataset, IQR is calculated as 53 - 44 = 9.
  • The IQR is less affected by outliers and skewed data than other measures like standard deviation.
Considering the IQR enables us to understand how spread out the middle half of the data is, providing insights into data consistency and variability.
Outliers Detection
Outliers are data points that differ significantly from other observations. They can indicate variability in the data or errors. To identify outliers, use the IQR:
  • Outliers are values lower than \( Q1 - 1.5 \times \text{IQR} \) or higher than \( Q3 + 1.5 \times \text{IQR} \).
  • For our dataset, this means looking for values below 30.5 or above 66.5.
  • In this case, there are no outliers as all data are within the range of 33 to 60.
Detecting outliers is crucial as they can skew the results of data analyses and affect the accuracy of statistical models. Understanding their presence helps in making informed adjustments or interpretations.

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Most popular questions from this chapter

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The San Luis ObispoTelegram-Tribune(October1,1994) reported the following monthly salaries for supervisors from six different counties: \(\$ 5,354\) (Kern), \(\$ 5,166\) (Monterey), \(\$ 4,443\) (Santa Cruz), \(\$ 4,129\) (Santa Barbara), \(\$ 2,500\) (Placer), and \$2,220 (Merced). San Luis Obispo County supervisors are supposed to be paid the average of the two counties in the middle of this salary range. Which measure of center determines this salary, and what is its value? Find the value of the other measure of center featured in this chapter. Why is it not as favorable to the San Luis Obispo County supervisors (although it might appeal to taxpayers)?

The following data values are the 2009 per capita expenditures on public libraries for each of the \(50 \mathrm{U} . \mathrm{S}\). states and Washington, D.C. (from www.statemaster.com): \(\begin{array}{rrrrrrr}16.84 & 16.17 & 11.74 & 11.11 & 8.65 & 7.69 & 7.48 \\\ 7.03 & 6.20 & 6.20 & 5.95 & 5.72 & 5.61 & 5.47 \\ 5.43 & 5.33 & 4.84 & 4.63 & 4.59 & 4.58 & 3.92 \\ 3.81 & 3.75 & 3.74 & 3.67 & 3.40 & 3.35 & 3.29 \\ 3.18 & 3.16 & 2.91 & 2.78 & 2.61 & 2.58 & 2.45 \\ 2.30 & 2.19 & 2.06 & 1.78 & 1.54 & 1.31 & 1.26 \\ 1.20 & 1.19 & 1.09 & 0.70 & 0.66 & 0.54 & 0.49 \\ 0.30 & 0.01 & & & & & \end{array}\) a. Summarize this data set with a frequency distribution. Construct the corresponding histogram. b. Use the histogram from Part (a) to find the approximate values of the following percentiles: i. 50 th iv. 90 th ii. 70 th v. 40 th iii. 10 th
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