Problems — 8.12 Analysing Data | Mathematics Department

1

**Reaction-distance dataset (department Numberphile).** A class of 20 Year 8 students measured reaction distance (cm): 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 19, 20, 20, 21, 23, 28. (a) Find the five-number summary: min, Q1, median, Q3, max. (b) Find the IQR. (c) Sketch a boxplot. (d) Using the 1.5×IQR rule, identify outliers.

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2

**Outlier sensitivity.** The dataset has mean 16.25 and median 15.5. (a) Remove the largest value (28). Find the new mean and median. (b) Compare the changes. (c) Which measure is more robust to outliers? Explain.

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3

**Comparing two boxplots.** Class A and Class B reaction distances: - Class A: min 8, Q1 12, median 15, Q3 18, max 25. - Class B: min 10, Q1 13, median 16, Q3 22, max 30. (a) Which class has a higher median? (b) Which class is more consistent (smaller IQR)? (c) Describe the skew of each. (d) Make a one-sentence comparison.

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4

**Combining two classes.** Class A: 20 students, mean 16.25. Class B: 30 students, mean 14.50. (a) Find the combined mean. (b) Why is this NOT the average of the two means?

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5

**Skew investigation.** A distribution of salaries in a small company has mean £45 000 and median £32 000. (a) Identify the skew direction. (b) What does this tell you about the salary distribution? (c) The CEO's salary is £200 000. Is this an outlier? Justify. (d) Which measure of central tendency is more representative of a typical employee?

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6

**Sampling-based inference.** A sample of 30 students has mean 75 and SD 10. The school of 600 students is the population. (a) Estimate the population mean. (b) Comment on the reliability with $n = 30$. (c) Estimate the percentage of students within ±10 marks of 75 (one SD). (d) Estimate the percentage between 65 and 85.

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7

**Inference from a small sample.** A poll of 50 people finds 30 prefer brand A. Estimate the percentage preference in the population, and state your confidence.

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8

**Comparing two data sets.** Class A and Class B test scores: A: 60, 70, 75, 80, 85. B: 50, 60, 75, 80, 100. (a) Find mean, median, range for each. (b) Which has higher mean? Higher median? (c) Which has more spread? (d) Comment on skew.

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9

**Inference validity.** A school claims its students have an average grade of 75. To verify: (a) Suggest two ways the school could test this claim. (b) State two issues that might cause the published average to be unreliable. (c) If sampling 50 students, what would be an acceptable margin of error?

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10

**MAD calculation.** Compute the Mean Absolute Deviation (MAD) of the data set 5, 8, 10, 12, 15. (a) Find the mean. (b) Find absolute deviations. (c) Find the MAD. (d) Interpret what MAD tells us.

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11

**Box-and-whisker investigation.** 30 students' weekly reading hours have 5-number summary: 1, 3, 6, 10, 25. (a) Find IQR. (b) Identify any outliers using the 1.5×IQR rule. (c) Sketch the boxplot and describe the skew. (d) What might the value 25 represent?

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12

**Validating an inference.** A teacher claims: "students who attend tutorials get higher marks". To support this: (a) Suggest a data-collection method. (b) State what statistical measures would help test the claim. (c) List two confounding factors that could mislead the conclusion. (d) Describe how a control group would help.

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