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Categorical Data

Grade 11 · Statistics · Worksheet 2

  1. A medical researcher is studying the effectiveness of a new drug for reducing cholesterol levels. In a clinical trial with 180 patients, the mean reduction in LDL cholesterol was 28.4 mg/dL with a standard deviation of 7.2 mg/dL. The researcher wants to test if this reduction is significantly greater than the standard treatment's average reduction of 25 mg/dL. Using a significance level of α = 0.01, calculate the test statistic and determine whether there is sufficient evidence to support the claim that the new drug provides a greater reduction than the standard treatment.
    • A. z = 4.24, Reject H₀
    • B. z = 2.12, Fail to reject H₀
    • C. z = 6.36, Reject H₀
    • D. z = 3.18, Fail to reject H₀
  2. A circle is inscribed in a right triangle with legs measuring 8 cm and 15 cm. The circle is tangent to all three sides of the triangle. What is the radius of the inscribed circle? Answer: ______________
  3. Hana surveyed 36 students about their preferred learning method: 14 prefer visual, 12 prefer auditory, 10 prefer kinesthetic. Create a frequency table summarizing this categorical data. Answer: ______________
  4. Isabella surveyed 27 students about their favorite subject. The results were: 12 Math, 7 Science, 2 History, 6 English. Create a frequency table showing the counts for each subject. Answer: ______________
  5. Isabella surveyed 72 students about their favorite subject and preferred learning style. The results showed: 17 students like Math and Visual, 12 like Math and Auditory, 22 like Science and Visual, 7 like Science and Auditory, 9 like English and Visual, and 5 like English and Auditory. Create a two-way frequency table summarizing this categorical data. Answer: ______________
  6. Aroha surveyed 63 students about their preferred learning environment. 21 preferred classroom, 17 preferred online, 13 preferred hybrid, and 12 preferred outdoor. Create a frequency table summarizing this categorical data. Answer: ______________
  7. Emma surveyed students about their favorite subject: Math: 17, Science: 13, English: 9, History: 11. Create a frequency table summarizing the data. Answer: ______________
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Answer Key & Explanations

Categorical Data · Grade 11 · Worksheet 2

  1. A medical researcher is studying the effectiveness of a new drug for reducing cholesterol levels. In a clinical trial with 180 patients, the mean reduction in LDL cholesterol was 28.4 mg/dL with a standard deviation of 7.2 mg/dL. The researcher wants to test if this reduction is significantly greater than the standard treatment's average reduction of 25 mg/dL. Using a significance level of α = 0.01, calculate the test statistic and determine whether there is sufficient evidence to support the claim that the new drug provides a greater reduction than the standard treatment. Answer: C. z = 6.36, Reject H₀ Solution: H₀: μ ≤ 25 mg/dL (new drug is not better than standard treatment) H₁: μ > 25 mg/dL (new drug provides greater reduction) z = (x̄ - μ) / (σ/√n) z = (28.4 - 25) / (7.2/√180) z = 3.4 / (7.2/13.4164) z = 3.4 / 0.5367 z = 6.36 For α = 0.01 in a right-tailed test, the critical z-value is 2.33 Since…
    Full step-by-step solution

    Step 1: State the hypotheses H₀: μ ≤ 25 mg/dL (new drug is not better than standard treatment) H₁: μ > 25 mg/dL (new drug provides greater reduction) Step 2: Calculate the test statistic z = (x̄ - μ) / (σ/√n) z = (28.4 - 25) / (7.2/√180) z = 3.4 / (7.2/13.4164) z = 3.4 / 0.5367 z = 6.36 Step 3: Determine the critical value For α = 0.01 in a right-tailed test, the critical z-value is 2.33 Step 4: Make a decision Since 6.36 > 2.33, we reject the null hypothesis Step 5: State the conclusion There is sufficient evidence at the 0.01 significance level to support the claim that the new drug provides a greater reduction in LDL cholesterol than the standard treatment. The correct answer is z = 6.36, Reject H₀.

  2. A circle is inscribed in a right triangle with legs measuring 8 cm and 15 cm. The circle is tangent to all three sides of the triangle. What is the radius of the inscribed circle? Answer: 3 Solution: Calculate the hypotenuse of the right triangle using the Pythagorean theorem. Hypotenuse = sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17 cm. Calculate the semiperimeter (s) of the triangle.
    Full step-by-step solution

    Step 1: Calculate the hypotenuse of the right triangle using the Pythagorean theorem. Hypotenuse = sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17 cm. Step 2: Calculate the semiperimeter (s) of the triangle. s = (8 + 15 + 17) / 2 = 40 / 2 = 20 cm. Step 3: Calculate the area (A) of the triangle. A = (1/2) * leg1 * leg2 = (1/2) * 8 * 15 = 60 cm². Step 4: Use the formula for the inradius (r) of a triangle: r = A / s. r = 60 / 20 = 3 cm. The radius of the inscribed circle is 3 cm.

  3. Hana surveyed 36 students about their preferred learning method: 14 prefer visual, 12 prefer auditory, 10 prefer kinesthetic. Create a frequency table summarizing this categorical data. Answer: Visual: 14, Auditory: 12, Kinesthetic: 10, Total: 36 Solution: Identify the categories: Visual, Auditory, Kinesthetic Record the frequency for each category: Visual = 14, Auditory = 12, Kinesthetic = 10 Calculate the total: 14 + 12 + 10 = 36 Visual | 14 Auditory | 12 Kinesthetic | 10 Total | 36
    Full step-by-step solution

    Step 1: Identify the categories: Visual, Auditory, Kinesthetic Step 2: Record the frequency for each category: Visual = 14, Auditory = 12, Kinesthetic = 10 Step 3: Calculate the total: 14 + 12 + 10 = 36 Step 4: Construct the frequency table: Preferred Learning Method | Frequency Visual | 14 Auditory | 12 Kinesthetic | 10 Total | 36

  4. Isabella surveyed 27 students about their favorite subject. The results were: 12 Math, 7 Science, 2 History, 6 English. Create a frequency table showing the counts for each subject. Answer: Math: 12, Science: 7, History: 2, English: 6 Solution: Step 1: Identify the categories: Math, Science, History, English Step 2: Record the count for each category from the survey data Step 3: Math: 12 students Step 4: Science: 7 students Step 5: History: 2 students Step 6: English: 6 students Step 7: Verify the total: 12 + 7 + 2 + 6 = 27 students…
    Full step-by-step solution

    Step 1: Identify the categories: Math, Science, History, English Step 2: Record the count for each category from the survey data Step 3: Math: 12 students Step 4: Science: 7 students Step 5: History: 2 students Step 6: English: 6 students Step 7: Verify the total: 12 + 7 + 2 + 6 = 27 students The frequency table shows: Math: 12, Science: 7, History: 2, English: 6

  5. Isabella surveyed 72 students about their favorite subject and preferred learning style. The results showed: 17 students like Math and Visual, 12 like Math and Auditory, 22 like Science and Visual, 7 like Science and Auditory, 9 like English and Visual, and 5 like English and Auditory. Create a two-way frequency table summarizing this categorical data. Answer: 72 Solution: Set up the two-way table structure with Subjects (Math, Science, English) as rows and Learning Styles (Visual, Auditory) as columns, plus row and column totals. - Math & Visual: 17 - Math & Auditory: 12 - Science & Visual: 22 - Science & Auditory: 7 - English & Visual: 9 - English & Auditory: 5…
    Full step-by-step solution

    Step 1: Set up the two-way table structure with Subjects (Math, Science, English) as rows and Learning Styles (Visual, Auditory) as columns, plus row and column totals. Step 2: Fill in the given frequencies: - Math & Visual: 17 - Math & Auditory: 12 - Science & Visual: 22 - Science & Auditory: 7 - English & Visual: 9 - English & Auditory: 5 Step 3: Calculate row totals: - Math total: 17 + 12 = 29 - Science total: 22 + 7 = 29 - English total: 9 + 5 = 14 Step 4: Calculate column totals: - Visual total: 17 + 22 + 9 = 48 - Auditory total: 12 + 7 + 5 = 24 Step 5: Calculate grand total from row totals: 29 + 29 + 14 = 72 Or from column totals: 48 + 24 = 72 The completed two-way frequency table is: Visual Auditory Total Math 17 12 29 Science 22 7 29 English 9 5 14 Total 48 24 72 The grand total is 72 students.

  6. Aroha surveyed 63 students about their preferred learning environment. 21 preferred classroom, 17 preferred online, 13 preferred hybrid, and 12 preferred outdoor. Create a frequency table summarizing this categorical data. Answer: Classroom: 21, Online: 17, Hybrid: 13, Outdoor: 12 Solution: List all categories: Classroom, Online, Hybrid, Outdoor. Classroom: 21 Online: 17 Hybrid: 13 Outdoor: 12
    Full step-by-step solution

    Step 1: List all categories: Classroom, Online, Hybrid, Outdoor. Step 2: Record the frequency for each category: - Classroom: 21 students - Online: 17 students - Hybrid: 13 students - Outdoor: 12 students Step 3: Verify the total: 21 + 17 + 13 + 12 = 63 students. Step 4: The completed frequency table is: Classroom: 21 Online: 17 Hybrid: 13 Outdoor: 12

  7. Emma surveyed students about their favorite subject: Math: 17, Science: 13, English: 9, History: 11. Create a frequency table summarizing the data. Answer: Math: 17, Science: 13, English: 9, History: 11 Solution: Identify the categories: Math, Science, English, History. List the corresponding frequencies: 17, 13, 9, 11.
    Full step-by-step solution

    Step 1: Identify the categories: Math, Science, English, History. Step 2: List the corresponding frequencies: 17, 13, 9, 11. Step 3: Construct the frequency table: | Subject | Frequency | |----------|-----------| | Math | 17 | | Science | 13 | | English | 9 | | History | 11 | The summarized data is Math: 17, Science: 13, English: 9, History: 11.