Question

questions on descriptive statistics
the next five questions refer to the following numbers in a dataset:
62, 12, 88, 27, 95, 24, 34, 55, 18, 51, 81, 77

question 21 (1 point)
the median is:
○ 51
○ 53
○ 55
○ 49

question 22 (1 point)
the mean is:
○ 52
○ 56
○ 50
○ 54

question 23 (1 point)
q1 falls in the:
○ 55,62 interval
○ 34,51 interval
○ 12,18 interval
○ 24,27 interval

question 24 (1 point)
the mode is:
○ 24
○ 51
○ 77
○ there is no mode

question 25 (1 point)
q3 falls in the:
○ 88,95 interval
○ 27,34 interval
○ 51,55 interval
○ 77,81 interval

Explanation:

Question 21

Step1: Sort the dataset

First, we sort the given dataset: \(12, 18, 24, 27, 34, 51, 55, 62, 77, 81, 88, 95\) (Wait, wait, let's count the numbers. The original dataset is 62, 12, 88, 27, 95, 24, 34, 55, 18, 51, 81, 77. So that's 12 numbers. Let's sort them correctly: \(12, 18, 24, 27, 34, 51, 55, 62, 77, 81, 88, 95\). Wait, no, 12 numbers: positions 1 - 12. The median for an even - numbered dataset is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th terms, where \(n = 12\). So \(\frac{n}{2}=6\) and \(\frac{n}{2}+1 = 7\). The 6th term is 51 and the 7th term is 55? Wait, no, let's re - sort:

Original data: 62, 12, 88, 27, 95, 24, 34, 55, 18, 51, 81, 77.

Sorting in ascending order: 12, 18, 24, 27, 34, 51, 55, 62, 77, 81, 88, 95.

Now, \(n = 12\) (even). The median is the average of the \(\frac{12}{2}=6\)th and \((\frac{12}{2}+1)=7\)th values.

The 6th value is 51, the 7th value is 55. Then median \(=\frac{51 + 55}{2}=\frac{106}{2}=53\).

Step2: Calculate the median

Median \(=\frac{\text{6th term}+\text{7th term}}{2}=\frac{51 + 55}{2}=53\)

Step1: Sum all the numbers

Sum \(S=12 + 18+24 + 27+34 + 51+55 + 62+77 + 81+88 + 95\)

Let's calculate step - by - step:

\(12+18 = 30\); \(30+24 = 54\); \(54+27 = 81\); \(81+34 = 115\); \(115+51 = 166\); \(166+55 = 221\); \(221+62 = 283\); \(283+77 = 360\); \(360+81 = 441\); \(441+88 = 529\); \(529+95 = 624\)

Step2: Calculate the mean

The mean \(\bar{x}=\frac{S}{n}\), where \(n = 12\) (number of data points) and \(S = 624\). So \(\bar{x}=\frac{624}{12}=52\)? Wait, no, \(624\div12 = 52\)? Wait, 12×52 = 624. Wait, but let's check the sum again.

Wait, original data: 62, 12, 88, 27, 95, 24, 34, 55, 18, 51, 81, 77.

Let's add them in a different order:

\(62+12 = 74\); \(88+27 = 115\); \(95+24 = 119\); \(34+55 = 89\); \(18+51 = 69\); \(81+77 = 158\)

Now sum these results: \(74+115 = 189\); \(189+119 = 308\); \(308+89 = 397\); \(397+69 = 466\); \(466+158 = 624\). So the sum is 624. Mean \(=\frac{624}{12}=52\)? Wait, but the options have 52, 56, 50, 54. Wait, maybe I made a mistake. Wait, 12 numbers:

Wait, 12+18 = 30; 30+24 = 54; 54+27 = 81; 81+34 = 115; 115+51 = 166; 166+55 = 221; 221+62 = 283; 283+77 = 360; 360+81 = 441; 441+88 = 529; 529+95 = 624. 624 divided by 12 is 52. So the mean is 52.

Step2: Calculate the mean

Mean \(=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{624}{12}=52\)

Step1: Find the position of Q1

For a dataset with \(n = 12\) (even), the first quartile \(Q1\) is the median of the first half of the data. The first half of the data is the first \(\frac{n}{2}=6\) numbers: \(12, 18, 24, 27, 34, 51\)

Step2: Calculate Q1

The median of a dataset with \(n = 6\) (even) is the average of the \(\frac{6}{2}=3\)rd and \((\frac{6}{2}+1)=4\)th terms. The 3rd term is 24, the 4th term is 27. Wait, no, the first half is \(12, 18, 24, 27, 34, 51\). The median of these 6 numbers is \(\frac{24 + 27}{2}=\frac{51}{2}=25.5\). Now we need to see in which interval 25.5 lies. The intervals are \([55,62]\), \([34,51]\), \([12,18]\), \([24,27]\). 25.5 is between 24 and 27, so it falls in the \([24,27]\) interval? Wait, no, wait the first half: when \(n = 12\), the first quartile is the median of the first 6 numbers. Wait, the sorted data is \(12, 18, 24, 27, 34, 51, 55, 62, 77, 81, 88, 95\). The first quartile (Q1) is the median of the data below the median. The median is between the 6th and 7th terms (51 and 55). So the data below the median is the first 6 terms: \(12, 18, 24, 27, 34, 51\). The median of these 6 terms: the 3rd term is 24, the 4th term is 27. So Q1 \(=\frac{24 + 27}{2}=25.5\). Now, 25.5 is in the interval \([24,27]\) (since 24 ≤ 25.5 ≤ 27).

Step2: Determine the interval for Q1

Q1 is 25.5, which lies in \([24,27]\) interval.

Answer:

53

Question 22