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## Statistics and Probability

### Basic Statistics - Standard Deviation and Variance

#### Q.01

'Assume that the heights of 500 second-year male students in a high school follow a normal distribution with a mean of 170.1 cm and a standard deviation of 5.6 cm.'

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'A scored 57.2 with a standard deviation of 5.2, while B scored 52.5 with a standard deviation of 9.5, and both scored 66. Assuming that all test scores follow a normal distribution, answer the following questions: (1) By calculating the z-scores, determine which of A and B is ranked higher relative to the entire population. (2) Assuming there were 2000 test takers for each exam, confirm the conclusion of (1) by determining the approximate ranks of A and B from the top.'

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'Please find the value for u = 0.5 on the normal distribution table.'

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'(2) (ii) When the number of people is increased from 400 to 900 and scores are extracted, the standard deviation S of the sample is 9.8 points. The 95% confidence interval C≤m≤D for the population mean m obtained from this sample, compared to the confidence interval A≤m≤B in (i), the width of the range is? (Fill in the blank) The same(1) becomes narrower(2) becomes wider'

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'Please explain the method of calculating standard deviation.'

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"A nationwide standardized test is held annually, with a perfect score of 200 points, and those scoring 100 points or more are considered passing. For this year's test, only the average score of all test takers is disclosed to be 95 points, with a standard deviation of 20 points. Assuming that the distribution of scores among all test takers is normal, answer the following questions."

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'The following data lists the number of days without rainfall (or snowfall) in Sapporo and Naha for each month of a particular year. Determine the range of each dataset and compare the spread of the data.'

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'Among the following 0 to 3, choose the most appropriate one for determining the degree of variation in catch volume? (0) Mean (1) Median (2) Standard Deviation (3) Mode'

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"For data of variable x, let the mean be x̄ and the standard deviation be sx. When new data of variable u is obtained through u=ax+b (a, b are constants), let the standard deviation of u's data be su, then su = |a| sx holds true."

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'If the results of a 6-point test for 25 students are given in a frequency distribution table as shown in columns 1 and 2 of the table, calculate the variance.'

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'(4) The following (a)~(c) describe the standard deviation when data on the catch of two marine products are transformed.'

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"What should you do when you don't understand a standard example?"

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'Basic 152 formula for relationship between variance and mean'

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'[Figure 1] to [Figure 3] below are histograms of the maximum daily temperatures in August in Tokyo for the years 1955, 1985, and 2015.'

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'Calculate the variance of the data consisting of 10, 7, 8, 0, 4, 2. Round to the nearest second decimal place.'

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'The figures [Figure 1] [Figure 3] below show histograms of the highest daily temperatures in Tokyo in August of 1955, 1985, and 2015. [Figure 1] August 1955\n[Figure 2] August 1985\n[Figure 3] August 2015\nAssume that the variance of the data for 1955, 1985, and 2015 is included in the following 0 to numbers. In this case, the variance of the data for 1955 is A, the variance of the data for 1985 is Y, and the variance of the data for 2015 is WOW. Choose one option from the following 0 to numbers to fill in the blanks.'

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'Investigate the spread of scores by class in a certain test.'

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'Find the variance and standard deviation of the given set of 151 data points'

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'Problems related to deviation, variance, and standard deviation: Given the values of variable x as follows:\n1. Find the deviation\n2. Find the variance\n3. Find the standard deviation'

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'For two variables x and y, if the standard deviation of x is 1.2, the standard deviation of y is 2.5, and the covariance between x and y is 1.08, find the correlation coefficient between x and y.'

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'For 8 cities overseas, the approximate flight times x from Narita Airport were investigated, and the following data was obtained: 7, 5, 7, 6, 8, 7, 10, 6 (in hours). Calculate the variance and standard deviation of this data. Round off to two decimal places if necessary.'

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'In a school, a math test was conducted on 1000 students and the results followed a normal distribution with a mean of 48 points and a standard deviation of 15 points. What is the score of the 30th student from the top? Round to the nearest tenth.'

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'PRACTICE 68\nA certain product consisting of 10,000 pieces follows a normal distribution with an average length of 69 cm and a standard deviation of 0.4 cm. When products with a length of 70 cm or more are considered defective, what percentage of defective products is expected to be included in these 10,000 products?'

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'Difference between variance and mean. Data is less concentrated around the mean and more spread out away from the mean. --> High variance Data is more concentrated around the mean and less spread out away from the mean. --> Low variance Using a histogram, consider the situation when combining datasets with the same mean and when combining datasets with the same variance. When combining datasets with the same mean --> When combining datasets with the same variance The spread is between the two data --> The variance is between the two data Becomes more spread out --> Variance increases'

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'Factory A: Mean 3.90 g, Standard Deviation 0.17 g Factory B: Mean 4.00 g, Standard Deviation 0.11 g (2) Factory A has a higher degree of dispersion'

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'(2) Compare the degree of dispersion from the mean of the data based on the standard deviation for the data from both factories.'

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'Translate the given text into multiple languages.'

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'The average of 185 is 55, with a standard deviation of 18'

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'Given variable data with a mean of 50 and a standard deviation of 15. Modify the data by increasing each value by 1.2 times and subtracting 5 to find the new mean and standard deviation.'

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'Given that the mean of variable x, x̄=21, and the variance, s_{x}^{2}=12. Calculate the mean value, variance, and standard deviation of a new variable y obtained by the following equations: (1) y=x-5 (2) y=3x (3) y=-2x+3 (4) y=(x-21)/(2√3). Let a, b be constants. For new data variable y obtained from data variable x by y=ax+b, the mean values of x and y are x̄, ȳ, the variances are s_{x}^{2}, s_{y}^{2}, and the standard deviations are s_{x}, s_{y}, where (1) ȳ=a x̄+b (2) s_{y}^{2}=a^{2} s_{x}^{2} (3) s_{y}=|a| s_{x} hold. Calculate using these rules.'

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'The following (a)~(c) describe the standard deviation when data on the catch of two marine products are transformed. (a) When each data is divided by 1000, the standard deviation remains the same as the original data. (b) When each data is subtracted by the mean catch of that marine product, the standard deviation remains the same as the original data. (c) When each data is subtracted by the mean catch of that marine product and then divided by 1000, the standard deviation remains the same as the original data. Choose one correct combination of (a)~(c) as G. Choose one number from 0 to 7 to represent the correct combination.'

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'Given the frequency distribution table of the results of a 6-point test for 25 students as shown in columns 1 and 2, calculate the variance.'

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'Given that the average of three positive numbers a, b, and c is 14, with a standard deviation of 8, find the values of a^2+b^2+c^2 and ab+bc+ca.'

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'The degree of variation in catch is the degree of dispersion of catch data. Among the options, the standard deviation (σ) represents the degree of data dispersion. When the catch is stable, it means that the degree of data dispersion is small, that is, the value of the standard deviation is small. Since the standard deviation takes values greater than or equal to 0, the closer the standard deviation is to 0, the more stable the catch is.'

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'Explain the dispersion of data and quartiles, and provide the steps to calculate quartiles.'

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'How to differentiate between two discrete computational expressions?'

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'Variable Transformation and Mean, Variance, Standard Deviation'

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'Answer the following questions regarding the data of variables x and y:\n(1) Calculate the variance s_x^2 for variable x and the variance s_y^2 for variable y.\n(2) Compare the data of variables x and y in terms of the degree of dispersion from the mean using standard deviation.'

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'(1) x and y have average values of 5 and 6 respectively; variances of 2 and 5.2; and standard deviations of 1.4 and 2.3. (2) The data for y has a greater degree of dispersion from the mean.'

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'Let 244 (51 a, b, c) be three distinct positive integers. The following data summarizes the scores of 10 people who took exams for subjects X and Y.'

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'Please tell me about variance and standard deviation.'

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"The variance of u's data is \\( \\overline{u^{2}}-(\ar{u})^{2}=\\frac{78}{6}-\\left(\\frac{-12}{6}\\right)^{2}=9 \\) Therefore, the standard deviation of u's data is \\( \\sqrt{9}=3(\\mathrm{~m}) \\) So, the standard deviation of x's data is \\( 4 \\times 3=12(\\mathrm{~m}) \\) Therefore, the variance of x's data is \ 12^{2}=144 \"