Categorical Data

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Answer Key & Explanations

Categorical Data · Grade 11 · Worksheet 1

1. Liam surveyed 63 students about their preferred outdoor activity. 21 chose hiking, 17 chose cycling, 13 chose kayaking, and 12 chose rock climbing. Create a frequency table summarizing this categorical data. Answer: Hiking: 21, Cycling: 17, Kayaking: 13, Rock climbing: 12 Solution: Identify the categories: Hiking, Cycling, Kayaking, Rock climbing. Hiking: 21 Cycling: 17 Kayaking: 13 Rock climbing: 12
   Full step-by-step solution

   Step 1: Identify the categories: Hiking, Cycling, Kayaking, Rock climbing. Step 2: Record the frequency for each category: - Hiking: 21 students - Cycling: 17 students - Kayaking: 13 students - Rock climbing: 12 students Step 3: Verify the total: 21 + 17 + 13 + 12 = 63 students. Step 4: The completed frequency table is: Hiking: 21 Cycling: 17 Kayaking: 13 Rock climbing: 12

2. Sophia surveyed students about their favorite music genres and created this two-way table showing preferences by gender: Music Genre Preferences Rock Hip-Hop Classical Total Male 12 8 5 25 Female 7 11 9 27 Total 19 19 14 52 What is the frequency of female students who prefer Classical music? Answer: 9 Solution: Find the value at the intersection of the Female row and Classical column The table shows 9 female students prefer Classical music Therefore, the frequency is 9
   Full step-by-step solution

   Step 1: Identify the row for Female students in the table Step 2: Identify the column for Classical music Step 3: Find the value at the intersection of the Female row and Classical column Step 4: The table shows 9 female students prefer Classical music Step 5: Therefore, the frequency is 9

3. Mason surveyed 84 students about their favorite type of music and whether they play an instrument. The results are: 22 play an instrument and prefer Rock, 14 play an instrument and prefer Pop, 10 do not play an instrument and prefer Rock, 18 do not play an instrument and prefer Pop, 8 play an instrument and prefer Jazz, 12 do not play an instrument and prefer Jazz. Create a two-way frequency table summarizing this categorical data. Answer: 84 Solution: Set up the two-way table structure with rows for instrument status and columns for music genre. The completed two-way frequency table shows all frequencies and totals correctly sum to 84 students.
   Full step-by-step solution

   Step 1: Set up the two-way table structure with rows for instrument status and columns for music genre. Step 2: Fill in the given frequencies: - Plays Instrument & Rock: 22 - Plays Instrument & Pop: 14 - Plays Instrument & Jazz: 8 - Does Not Play Instrument & Rock: 10 - Does Not Play Instrument & Pop: 18 - Does Not Play Instrument & Jazz: 12 Step 3: Calculate row totals: - Plays Instrument total: 22 + 14 + 8 = 44 - Does Not Play Instrument total: 10 + 18 + 12 = 40 Step 4: Calculate column totals: - Rock total: 22 + 10 = 32 - Pop total: 14 + 18 = 32 - Jazz total: 8 + 12 = 20 Step 5: Verify grand total: 44 + 40 = 84, or 32 + 32 + 20 = 84. The completed two-way frequency table shows all frequencies and totals correctly sum to 84 students.

4. Emma surveyed 40 students at her school to determine their favorite type of music and their preferred method of listening to music (streaming or downloading). She found that 15 students preferred pop music, 10 preferred rock, and 15 preferred hip-hop. Of the pop music fans, 10 preferred streaming and 5 preferred downloading. Of the rock fans, 6 preferred streaming and 4 preferred downloading. Of the hip-hop fans, 12 preferred streaming and 3 preferred downloading. Create a two-way frequency table summarizing the data, showing the counts of students for each combination of music preference and listening method. Answer: A two-way frequency table with rows for Pop, Rock, Hip-hop and columns for Streaming, Downloading, and Total, showing the counts: Pop (10, 5, 15), Rock (6, 4, 10), Hip-hop (12, 3, 15), and totals (28, 12, 40). Solution: Identify the two categorical variables: music preference (pop, rock, hip-hop) and listening method (streaming, downloading). - Pop and streaming: 10 - Pop and downloading: 5 - Rock and streaming: 6 - Rock and downloading: 4 - Hip-hop and streaming: 12 - Hip-hop and downloading: 3 Calculate row…
   Full step-by-step solution

   Step 1: Identify the two categorical variables: music preference (pop, rock, hip-hop) and listening method (streaming, downloading). Step 2: Create a table with rows for each music type and columns for each listening method, plus a total column for rows and a total row for columns. Step 3: Fill in the cell counts based on the problem: - Pop and streaming: 10 - Pop and downloading: 5 - Rock and streaming: 6 - Rock and downloading: 4 - Hip-hop and streaming: 12 - Hip-hop and downloading: 3 Step 4: Calculate row totals: Pop total = 10 + 5 = 15; Rock total = 6 + 4 = 10; Hip-hop total = 12 + 3 = 15. Step 5: Calculate column totals: Streaming total = 10 + 6 + 12 = 28; Downloading total = 5 + 4 + 3 = 12. Step 6: Verify that the grand total (sum of row totals or column totals) equals 40: 15 + 10 + 15 = 40 and 28 + 12 = 40. The completed two-way frequency table is: | Streaming | Downloading | Total Pop | 10 | 5 | 15 Rock | 6 | 4 | 10 Hip-hop | 12 | 3 | 15 Total | 28 | 12 | 40

5. A pharmaceutical company is testing a new drug and claims it reduces recovery time by at least 48 hours. Researchers conduct a clinical trial with 120 patients and find a sample mean reduction of 45 hours with a standard deviation of 12 hours. Using a significance level of α = 0.05, test the hypothesis H₀: μ ≥ 48 against H₁: μ < 48. What is the calculated test statistic value? Answer: -2.74 Solution: - Sample size \( n = 120 \) - Sample mean \( \bar{x} = 45 \) - Population standard deviation is unknown; sample standard deviation \( s = 12 \) - Claimed population mean under null hypothesis \( \mu_0 = 48 \) - Significance level \( \alpha = 0.05 \) - Null hypothesis \( H_0: \mu \geq 48 \) -…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the given values** - Sample size \( n = 120 \) - Sample mean \( \bar{x} = 45 \) - Population standard deviation is unknown; sample standard deviation \( s = 12 \) - Claimed population mean under null hypothesis \( \mu_0 = 48 \) - Significance level \( \alpha = 0.05 \) - Null hypothesis \( H_0: \mu \geq 48 \) - Alternative hypothesis \( H_1: \mu < 48 \) This is a **one-tailed test** (left-tailed). --- **Step 2: Choose the test statistic** Since \( n > 30 \) but population standard deviation is unknown, we can use the **t-test** (though with large \( n \), t is close to z). The formula for the t-test statistic is: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] --- **Step 3: Plug in the numbers** \[ t = \frac{45 - 48}{12 / \sqrt{120}} \] First, compute the denominator: \[ \sqrt{120} \approx 10.954451 \] \[ s / \sqrt{n} = 12 / 10.954451 \approx 1.095445 \] Now compute the numerator: \[ 45 - 48 = -3 \] So: \[ t = \frac{-3}{1.095445} \approx -2.738613 \] --- **Step 4: Round appropriately** The problem’s given correct answer is -2.74, so we round to two decimal places: \[ t \approx -2.74 \] --- **Step 5: Interpretation** The test statistic is -2.74, meaning the sample mean is 2.74 standard errors below the hypothesized population mean of 48. --- **Final Answer:** -2.74

6. Noah surveyed 56 students about their preferred type of music and their grade level. The results showed: 11th grade & Rock: 16, 11th grade & Pop: 11, 11th grade & Jazz: 6, 12th grade & Rock: 9, 12th grade & Pop: 8, 12th grade & Jazz: 6. Create a two-way frequency table summarizing this categorical data. Answer: 56 Solution: Set up the two-way table with Grade Level as rows (11th, 12th) and Music Genre as columns (Rock, Pop, Jazz).
   Full step-by-step solution

   Step 1: Set up the two-way table with Grade Level as rows (11th, 12th) and Music Genre as columns (Rock, Pop, Jazz). Step 2: Fill in the given frequencies: - 11th grade & Rock: 16 - 11th grade & Pop: 11 - 11th grade & Jazz: 6 - 12th grade & Rock: 9 - 12th grade & Pop: 8 - 12th grade & Jazz: 6 Step 3: Calculate row totals: - 11th grade total: 16 + 11 + 6 = 33 - 12th grade total: 9 + 8 + 6 = 23 Step 4: Calculate column totals: - Rock total: 16 + 9 = 25 - Pop total: 11 + 8 = 19 - Jazz total: 6 + 6 = 12 Step 5: Compute the grand total: 33 + 23 = 56 (or 25 + 19 + 12 = 56) The completed two-way frequency table: Rock Pop Jazz Total 11th Grade 16 11 6 33 12th Grade 9 8 6 23 Total 25 19 12 56 The grand total is 56 students.