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## Statistics and Probability

### Basic Statistics - Organizing and Displaying Data (Tables, Graphs)

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'Please explain the characteristics of the Red Chart series.'

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'(1) The expected value of X (2) The variance of X (3) The standard deviation of X When two dice are thrown simultaneously, there are 36 possible outcomes for the sum. The possible values of X are 1, 2, 3, 4, 5, 6. X=k (k is an integer such that 1 ≤ k ≤ 6) when one die shows k and the other shows a number greater than or equal to k but less than or equal to 6. Therefore, the probability distribution of X is as shown in the table on the right.'

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'Basic Example 40\nThere are 10, 5, 5, 5, and 10 balls with numbers 1, 2, 3, 4, 5 written on them, totaling 35 balls. Considering these 35 balls as the population and the numbers written on the balls as the variable X, find the population distribution, population mean m, and population standard deviation σ.'

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'For item 4 underlined section d, give one example of a situation where foreign tourists visiting Japan may have difficulty paying the fare when boarding a local bus. However, exclude issues related to communication problems with the driver and exchanging banknotes/coins. Please write your answer within the space provided on the answer sheet.'

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'There are three bottles of different gases, one each. It is unknown which bottle belongs to which gas. The names and characteristics of the three gases are as shown in Table 1. Carbon dioxide is also included for comparison. Operations 1 to 3 were carried out in the same place, at the same temperature.'

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'Analyze the presence of questions in each field of the test by year as shown in the table and identify the main areas.'

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"Question 11 For the underlined part k, the following table shows the trend of operating income of shipping companies. All companies have seen a significant increase in income from 1915. Considering the trade situation in Japan at that time, please answer within the answer sheet the reasons why the income of shipping companies grew during this period. | Fiscal Year Company Name | Nippon Yusen | Osaka Shosen | | 1913 | 3,403 | 2,018 | | 1914 | 3,419 | 1,935 | | 1915 | 4,210 | 2,360 | | 1916 | 6,819 | 4,367 | | 1917 | 11,604 | 7,246 | | 1918 | 22,291 | 16,787 | | 1919 | 21,676 | 12,717 | *Unit in 10,000 yen, created from 'Meiji Taisho Kokusei Soran' (Toyo Keizai Shinpo, 1927)"

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'Based on the graph below with the underline at 10 (i), explain in 25 words or less the background for the rise in rice prices.'

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'Explain frequency distribution table and histogram, and provide examples.'

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'Choose one appropriate scatter plot for this data from the following figures (A) to (C).'

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'Basic Example 146 Reading from a box plot\nThe figure on the right is a box plot of the sales data of products A and product B at a certain store over 30 days. Select all the correct statements that can be read from this box plot from the following (1) to (3).'

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'The diagram on the right shows the score distribution of 30 students in a science test in the form of a box plot. When creating a histogram based on this box plot, which of the following options should be chosen?'

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'For the data of variable y, calculate the number of outliers.'

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'The following data represents the distances between stations of a certain railway line: 1.99, 2.21, 1.45, 1.23, 1.05, 0.79, 1.28, 0.79, 0.95, 0.59, 1.09, 1.11, 0.52, 0.76, 1.57, 0.72, 1.12, 1.81, 1.22, 0.86, 1.42, 1.32, 0.74, 1.46, 1.21, 1.61, 1.50, 1.19, 0.89 (unit is in km).\n(1) Create a frequency distribution table with a class width of 0.2 km. Start dividing the classes from 0.4 km.\n(2) Create a histogram based on the frequency distribution table created in (1).'

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"The following data represents A's commuting time over 15 days (in minutes): 51, 49, 52, 54, 48, 55, 50, 51, 53, 46, 57, 54, 58, 52, 50. Choose the box plot that represents this data from options (1) - (3)."

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'The following data represents the record of the highest temperatures in city A for a certain month: 20.7, 20.1, 14.5, 10.9, 12.1, 19.1, 16.3, 13.1, 14.6, 20.2, 23.2, 14.3, 20.1, 17.4, 11.2, 7.4, 11.5, 16.5, 19.1, 18.1, 25.5, 14.2, 10.1, 16.7, 19.7, 19.1, 17.5, 23.4, 20.4, 10.1 (unit in °C). (1) Create a frequency distribution table with a class width of 2°C, starting from 6°C for the class intervals. (2) Based on the frequency distribution table created in (1), draw a histogram.'

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'The following data shows the distances between stations of a railway.'

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'Explain the types and features of statistical graphs, and provide examples of the characteristics and drawbacks of each.'

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"Let's reflect on the values represented by a box plot!"

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'Calculate the first quartile, second quartile, and third quartile of the data for A.'

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'The table below shows data on the list price and sales quantity of 10 types of bread.'

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'The following data is the stroke count of 30 last names in a certain high school class:\n\egin{tabular}{|rrrrrrrrrrrrrrrl}\n18 & 9 & 19 & 15 & 10 & 15 & 8 & 17 & 11 & 21 & 9 & 26 & 23 & 13 & 31 & \\\n14 & 9 & 27 & 10 & 11 & 9 & 17 & 18 & 19 & 11 & 6 & 12 & 15 & 18 & 11 & (strokes) \\\n\\hline\n\\end{tabular}\n(1) Create a frequency distribution table with a class width of 5 strokes, where one class includes strokes equal to or more than 5 and less than 10.\n(2) Based on the frequency distribution table from (1), draw a histogram.\n(3) How many students have last names with a stroke count of 20 or less?'

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'Investigate the monthly sales changes of product A.'

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'The figure to the right is a box plot of the scores of 30 students on Test A and Test B. Which of the following statements can be correctly inferred from this box plot? Select all that apply from (1) to (3).'

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'The following data is the sales quantity of Product A in a certain region for 30 days.'

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'The right figure is a box plot of the sales data of product A and product B in a certain store over 30 days. Which of the following can be correctly inferred from this box plot? (Select all that apply from 1 to 3).'

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"Let's reflect on the values represented in a box plot!"

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'Draw a basic scatter plot and examine the correlation.'

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'Investigate the proportion of responses in a multiple-choice survey.'

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'As a criterion for outliers, you can use the following: a value below the first quartile minus 1.5 times the interquartile range, and a value above the third quartile plus 1.5 times the interquartile range.'

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'In 2015, which type of scatter plot represents the relationship between the proportion of primary industry workers by prefecture (horizontal axis) and the proportion of female workers (vertical axis)? Please select the most appropriate answer.'

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'Show how to calculate the first and third quartiles for population data of six cities: 10, 24, 26, 28, 40, 370 million people, and demonstrate that 370 million people is an outlier.'

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'Given cards with numbers 1, 2, 3, 4, 5 written on them in quantities of 1, 2, 3, 4, 5 respectively totaling 15 cards. Consider this as the population, with the numbers on the cards as variable X. Determine the population distribution, population mean m, and population standard deviation σ.'

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'The table on the right shows the frequency distribution of test scores for 8 students. Assume all scores are integers.\n(1) Determine the range of possible average values for this data.\n(2) The average score for the 8 students is 52, with individual scores of 34, 42, 43, 46, 57, 58, 65, and x (in points). Find the value of x.\n\nScore Range (points) | Number of Students\n20 and above, less than 40 | 1\n40 and above, less than 60 | 5\n60 and above, less than 80 | 2\nTotal | 8'

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'Data may contain values that are significantly different from other values, known as outliers. There are multiple criteria to determine outliers, for example, considering values that are:\nBelow or equal to the first quartile minus 1.5 times the interquartile range\nAbove or equal to the third quartile plus 1.5 times the interquartile range\n\nFor the data set: 5,8,8,9,10,12,15,99\nThe first quartile is 8, the third quartile is 13.5, so the interquartile range is 13.5-8=5.5. Therefore,\n\n(Third quartile)+1.5*interquartile range\n=13.5+1.5*5.5=21.75\n\nHence, the value 99, which is above 21.75, can be considered as an outlier.'

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'The diagram on the right is a box plot of the test scores for 160 students in Mathematics I and Mathematics A. Choose all the correct statements from (1) to (4) based on this box plot.'

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'[Table 2] The cumulative frequency is 10, 10+15 (=25), 10+15+36 (=61), 10+15+36+34 (=95), 10+15+36+34+6 (=101), 10+15+36+34+6+4 (=105).'

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'The table summarizes the monthly new car registration numbers for passenger cars from 2008 to 2013. What can be analyzed from the data in this table?'

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'When the weight of 105 eggs (in grams) is measured with a class width of 5g as shown in Table 1, the cumulative frequency distribution table is as in Table 2, and the relative frequency distribution table is as in Table 3.'

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'The records of handball throws were re-recorded in this class the other day. A~D shown below describe the analysis results of the changes from the initial records to the current ones. If each of a~d represents a box plot of the re-recorded data, choose two out of the combinations of 10~13 that contradict the analysis results and box plots.'

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'What pages are found at the beginning of each chapter in the blue chart?'

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'The relative frequencies in Table 3 are 10/105 (≈0.10), 15/105 (≈0.14), 36/105 (≈0.34), 34/105 (≈0.32), 6/105 (≈0.06), 4/105 (≈0.04).'

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'Choose the appropriate scatter plot from (1) to (4) with attention to correlation and median.'

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'Data analysis\nMeasures of central tendency\nMean \\( \ar{x} \\quad \ar{x}=\\frac{1}{n}(x_{1}+x_{2}+\\cdots+x_{n}) \\)\nMedian\nThe middle value when the data is sorted in ascending order. If the data set has an even number of values, the median is the average of the two middle values.\nMode\nThe value that appears most frequently in a data set. In a frequency distribution table, it is the value of the class interval with the highest frequency.\nBox plot\nA graphical representation of the minimum value, first quartile \ Q_{1} \, median, third quartile \ Q_{3} \, and maximum value of a dataset using a box and whiskers.\nVariance and standard deviation\nDeviation The difference between each value of variable x and the mean\n\\nx_{1}-\\overline{x}, x_{2}-\\overline{x}, \\cdots , x_{n}-\\overline{x}\n\\nVariance The average of the squared deviations\n\\[\ns^{2}=\\frac{1}{n}\\{(x_{1}-\ar{x})^{2}+(x_{2}-\ar{x})^{2}+\\cdots+(x_{n}-\ar{x})^{2}\\}\n\\]\nStandard deviation The positive square root of the variance \ s=\\sqrt{\\text { variance }} \\nRelationship between variance and mean \\( s^{2}=\\overline{x^{2}}-(\ar{x})^{2} \\)\nCorrelation coefficient\nIf the standard deviations of variables x and y are \ s_{x}, s_{y} \ respectively, and the covariance of x and y is \ S_{xy} \, then the correlation coefficient \ r \ is\n\\[r=\\frac{s_{xy}}{s_{x} s_{y}}(-1 \\leqq r \\leqq 1)\\]\nHypothesis testing\nA method of making decisions about a population based on data obtained, by formulating a hypothesis and determining if it is true.'

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'The following data is the weight of 30 players on a baseball team.'

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'Which type of chart is best suited to examine the changes in the employed population over the past 20 years?'

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'Frequency Distribution Table and Measures of Central Tendency'

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'For a class of 40 third-year high school students, data on handball throwing distances was collected. The diagram on the right summarizes the data initially collected in a box-and-whisker plot. Subsequently, the class retook records of handball throwing. The following A to D describe the analysis results of the change from the initial records to the new records. For each of a to d, select all choices that contradict the analysis results of the box-and-whisker plot of the data retaken this time.'

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'Given data for two variables x and y. Draw a scatter plot and determine if there is a correlation between x and y. If there is a correlation, specify whether it is positive or negative.'

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'Basic Example 142 Frequency Distribution Table, Histogram\nThe following data is the record of the highest temperatures in city A for a certain month.\n(20.7, 20.1, 14.5, 10.9, 12.1, 19.1, 16.3, 13.1, 14.6, 20.2, 25.5, 14.2, 10.1, 16.7, 16.7, 19.9, 15.7, 15.4, 23.4, 20.1) (in °C)\n(1) Create a frequency distribution table with a class width of 2°C. The classes should start from 6°C.\n(2) Based on the frequency distribution table created in (1), draw a histogram.'

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'To investigate the relationship between math and science test results, which graph is most appropriate?'

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'Basic Example 146 Reading from Boxplot\nThe figure on the right is a boxplot of sales data for products A and B at a certain store over 30 days. From this boxplot, select all the correct statements from 1 to 3.\n(1) Product A has a larger range and interquartile range of sales compared to product B.\n(2) Product A had 15 or more days with sales of 15 items or more.\n(3) Both product A and B had days with less than 10 items sold.\nProduct A Product B'

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'Which graph is suitable for examining the spread of heights of 30 students?'

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'The following data is the distance between stations of a railway.'

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'How much does it cost for Chuasiki Basic Mathematics I + B (Blue Chart Mathematics I + B)?'

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'Describe the characteristics of the Blue Chart series.'

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'Introduced the red chart, blue chart, yellow chart, and white chart series. Please briefly explain the main characteristics of each series.'

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'What is the price of Chiyasu Keisoku Solution and Exercise Mathematics I+A (Yellow Chart Mathematics I+A)?'

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'Draw a scatter plot showing a positive correlation.'

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'The chart on the right shows a box plot of scores for 50 students in a school for 4 types of tests A, B, C, D. Each test is out of 100 points.'

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'The following data shows the sales quantity of products A and B in a store over 10 days (unit is in pieces).'

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'Create a histogram based on the following frequency distribution table. Data: [1-3], [4-6], [7-9]. There are 3, 5, and 2 data points in each respective range.'

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'Assuming n pairs of values are given as data consisting of two variables x and y: (x_{1}, y_{1}), (x_{2}, y_{2}), ... , (x_{n}, y_{n}). (1) Scatter plot. A scatter plot is created by taking these points as coordinates—(x_{1}, y_{1}), (x_{2}, y_{2}), ... , (x_{n}, y_{n})—on a coordinate plane to visually represent the relationship between x and y as shown in the right figure.'

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What should you do if you don't understand basic examples?

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Explain the connection between the question and the basic matters based on the chart-style editing policy.