crm.dsr.6.1 univariate data: represent with graphs on a number lines
if you were to create a histogram with 6 bars starting at 15 and ending at 74, how many bars would have a frequency of 3?
stem | leaf
1 | 6 8 8
2 | 1 7 9
3 | 0 3 4 9
4 | 1 4 4 5 5 8
5 | 4 5 7 9
6 | 1 3 6
7 | 0 2

Question

Accepted Answer

# Explanation: ## Step1: Calculate the class width The range of the data is $ 74 - 15 = 59 $. We have 6 bars, so the class width $ w $ is $ \frac{74 - 15}{6}=\frac{59}{6}\approx10 $ (we can also calculate it as $ \lceil\frac{74 - 15}{6} ceil = 10 $ or by checking the interval). Let's define the intervals: - First interval: $ 15 - 24 $ (since starting at 15, width 10: 15 - 24) - Second: $ 25 - 34 $ - Third: $ 35 - 44 $ - Fourth: $ 45 - 54 $ - Fifth: $ 55 - 64 $ - Sixth: $ 65 - 74 $ ## Step2: Count data in each interval - Interval 15 - 24: Data from stem 1 (16,18,18) and stem 2 (21,27,29)? Wait, no. Wait stem - leaf: stem is tens place, leaf is ones. So: - 15 - 24: numbers with tens digit 1 or 2, and ones digit such that number is between 15 - 24. So stem 1: 16,18,18 (all ≥15, ≤24). Stem 2: 21,27,29? Wait 27 and 29 are >24. So stem 2: 21 is in 15 - 24? 21 is between 15 - 24. 27 is 27, which is in 25 - 34. So 15 - 24: stem 1 (16,18,18) and stem 2 (21). Wait no: 15 - 24: numbers from 15 to 24 inclusive. So stem 1: 16,18,18 (all 16,18,18 are between 15 -24). Stem 2: 21 (21 is between 15 -24), 27 is 27 (25 -34), 29 is 29 (25 -34). So 15 -24: 16,18,18,21 → wait no, 21 is 21, which is in 15 -24? Yes, 21 is between 15 and 24. Wait 15 -24: 15 ≤ x <25? Wait histogram intervals are usually left - inclusive, right - exclusive. So 15 -24 (15 ≤ x <25), 25 -34 (25 ≤ x <35), etc. So: - 15 -24: stem 1 (16,18,18) and stem 2 (21,27? No, 27 ≥25). So stem 1: 16,18,18 (3 numbers). Stem 2: 21 (1 number). Wait no, 21 is 21, which is in 15 -24. So total for 15 -24: 16,18,18,21? Wait 16,18,18 are 16,18,18 (3 numbers), 21 is 21 (1 number). Wait no, 21 is 21, which is in 15 -24. So 3 (stem 1) +1 (stem 2:21) =4? Wait no, maybe I messed up. Let's list all data points: - Stem 1: 16,18,18 → 16,18,18 (3 numbers) - Stem 2: 21,27,29 → 21,27,29 (3 numbers) - Stem 3: 30,33,34,39 → 30,33,34,39 (4 numbers) - Stem 4: 41,44,44,45,45,48 → 41,44,44,45,45,48 (6 numbers) - Stem 5: 54,55,57,59 → 54,55,57,59 (4 numbers) - Stem 6: 61,63,66 → 61,63,66 (3 numbers) - Stem 7: 70,72 → 70,72 (2 numbers) Now assign to intervals: - 15 -24: numbers between 15 -24. So stem 1 (16,18,18) and stem 2 (21) → 16,18,18,21? Wait 21 is 21, which is in 15 -24. 27 and 29 are in 25 -34. So 15 -24: 16,18,18,21 → 4 numbers? Wait no, 16,18,18 are 3 numbers (stem 1), 21 is 1 number (stem 2) → total 4? Wait maybe my interval definition is wrong. Let's use width 10, starting at 15: 15 -24, 25 -34, 35 -44, 45 -54, 55 -64, 65 -74. Now: - 15 -24: numbers with tens digit 1 or 2, and number ≤24. So stem 1: 16,18,18 (all ≤24) → 3 numbers. Stem 2: 21 (≤24), 27 (>24), 29 (>24) → so stem 2 contributes 1. Total: 3 +1 =4? Wait no, 21 is 21, which is in 15 -24. So 16,18,18,21 → 4 numbers. - 25 -34: tens digit 2 or 3, number between 25 -34. Stem 2: 27,29 (27,29 are 27,29 which are 25 -34? 27 is 27 (25 -34), 29 is 29 (25 -34). Stem 3: 30,33,34 (30,33,34 are 30 -34, so 25 -34? Wait 30 is 30, which is in 25 -34? Yes, 25 -34 includes 30. So stem 2: 27,29 (2 numbers), stem 3: 30,33,34 (3 numbers) → total 2 +3 =5? Wait no, stem 3: 30,33,34,39. 39 is 39, which is in 35 -44. So stem 3: 30,33,34 (3 numbers) in 25 -34, 39 in 35 -44. So 25 -34: 27,29 (stem 2) and 30,33,34 (stem 3) → 2 +3 =5. - 35 -44: tens digit 3 or 4, number between 35 -44. Stem 3: 39 (39 is 35 -44? 39 is between 35 -44. Stem 4: 41,44 (41,44 are 35 -44? 41 is 41 (35 -44), 44 is 44 (35 -44). Stem 4: 45,45,48 are >44. So stem 3: 39 (1 number), stem 4: 41,44 (2 numbers) → total 1 +2 =3. - 45 -54: tens digit 4 or 5, number between 45 -54. Stem 4: 45,45,48 (45,45,48: 45 is 45 -54? 45 is 45, which is in 45 -54. 48 is 48 (45 -54). Stem 5: 54 (54 is 45 -54? 54 is 54, which is in 45 -54? 45 -54 includes 54? Wait 45 -54: 45 ≤x <55. So 54 is included. So stem 4: 45,45,48 (3 numbers), stem 5: 54 (1 number) → total 3 +1 =4? Wait no, 45,45,48 (stem 4) and 54 (stem 5) → 4 numbers. - 55 -64: tens digit 5 or 6, number between 55 -64. Stem 5: 55,57,59 (55,57,59: 55 is 55 -64? 55 is 55 (55 -64), 57,59 (55 -64). Stem 6: 61,63 (61,63 are 55 -64? 61 is 61 (55 -64), 63 is 63 (55 -64). Stem 6: 66 is 66 (65 -74). So stem 5: 55,57,59 (3 numbers), stem 6: 61,63 (2 numbers) → total 3 +2 =5. - 65 -74: tens digit 6 or 7, number between 65 -74. Stem 6: 66 (66 is 65 -74), stem 7: 70,72 (70,72 are 65 -74). So stem 6: 66 (1 number), stem 7: 70,72 (2 numbers) → total 1 +2 =3. Wait now let's re - check each interval: 1. 15 -24: - Stem 1: 16,18,18 (3 numbers) - Stem 2: 21 (1 number) - Total: 3 +1 =4. Wait but 21 is 21, which is in 15 -24. Correct. 2. 25 -34: - Stem 2: 27,29 (2 numbers) - Stem 3: 30,33,34 (3 numbers) - Total: 2 +3 =5. 3. 35 -44: - Stem 3: 39 (1 number) - Stem 4: 41,44 (2 numbers) - Total: 1 +2 =3. 4. 45 -54: - Stem 4: 45,45,48 (3 numbers) - Stem 5: 54 (1 number) - Total: 3 +1 =4. 5. 55 -64: - Stem 5: 55,57,59 (3 numbers) - Stem 6: 61,63 (2 numbers) - Total: 3 +2 =5. 6. 65 -74: - Stem 6: 66 (1 number) - Stem 7: 70,72 (2 numbers) - Total: 1 +2 =3. Now we need to find how many intervals (bars) have frequency 3. Looking at the counts: - 35 -44: 3 - 65 -74: 3 Wait wait, did I make a mistake in 15 -24? Wait stem 1: 16,18,18 (3 numbers). Stem 2: 21 (1 number). So 3 +1 =4. Correct. 25 -34: stem 2 (27,29) and stem 3 (30,33,34). 2 +3 =5. Correct. 35 -44: stem 3 (39) and stem 4 (41,44). 1 +2 =3. Correct. 45 -54: stem 4 (45,45,48) and stem 5 (54). 3 +1 =4. Correct. 55 -64: stem 5 (55,57,59) and stem 6 (61,63). 3 +2 =5. Correct. 65 -74: stem 6 (66) and stem 7 (70,72). 1 +2 =3. Correct. Wait also, let's check 15 -24 again. Wait the problem says "starting at 15 and ending at 74". Maybe the intervals are 15 -24, 25 -34, 35 -44, 45 -54, 55 -64, 65 -74 (6 intervals). Now, let's list all data points: From stem - leaf: 16,18,18, 21,27,29, 30,33,34,39, 41,44,44,45,45,48, 54,55,57,59, 61,63,66, 70,72. Now count each interval: 1. 15 -24: 16,18,18,21 → 4 numbers (frequency 4) 2. 25 -34: 27,29,30,33,34 → 5 numbers (frequency 5) 3. 35 -44: 39,41,44,44 → Wait wait! Oh no! I made a mistake earlier. Stem 4: 41,44,44,45,45,48. So 41,44,44 are in 35 -44? 41 is 41 (35 -44), 44 is 44 (35 -44), 44 is 44 (35 -44). Then stem 3: 39. So 35 -44: 39,41,44,44 → 4 numbers? Wait I messed up the stem - leaf. Stem is tens place, leaf is ones. So: - Stem 1 (tens =1): 16,18,18 → numbers 16,18,18 - Stem 2 (tens =2): 21,27,29 → numbers 21,27,29 - Stem 3 (tens =3): 30,33,34,39 → numbers 30,33,34,39 - Stem 4 (tens =4): 41,44,44,45,45,48 → numbers 41,44,44,45,45,48 - Stem 5 (tens =5): 54,55,57,59 → numbers 54,55,57,59 - Stem 6 (tens =6): 61,63,66 → numbers 61,63,66 - Stem 7 (tens =7): 70,72 → numbers 70,72 Now re - categorize: 1. 15 -24 (15 ≤ x <25): - Numbers with tens digit 1 or 2, and x <25. - Stem 1: 16,18,18 (all <25) → 3 - Stem 2: 21 (21 <25), 27 (27 ≥25), 29 (29 ≥25) → 1 - Total: 3 +1 =4 (correct as before) 2. 25 -34 (25 ≤ x <35): - Tens digit 2 or 3, x <35. - Stem 2: 27,29 (27,29 <35) → 2 - Stem 3: 30,33,34 (30,33,34 <35) → 3 - Total: 2 +3 =5 (correct) 3. 35 -44 (35 ≤ x <45): - Tens digit 3 or 4, x <45. - Stem 3: 39 (39 <45) → 1

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Question

Explanation:

Step1: Calculate the class width

Step2: Count data in each interval

Answer:

Step1: Calculate the class width

Step2: Count data in each interval