Solved on Feb 18, 2024

Find the third quartile for the set of lengths: 130, 170, $160, 160, 150, 190$ .

STEP 1

Assumptions
1. We are given a set of lengths: 130, 170, 160, 160, 150, 190.
2. We need to find the third quartile (Q3) of this data set.
3. The third quartile is the value below which 75% of the data falls.
4. The data must be ordered from smallest to largest before calculating the quartiles.

STEP 2

First, we will order the given lengths from smallest to largest.
$\text{Ordered lengths: } 130, 150, 160, 160, 170, 190$

STEP 3

To find the third quartile, we need to determine the position in the ordered list where 75% of the data lies below it. The position can be found using the formula:
$\text{Position of Q3} = \frac{3}{4}(n + 1)$
where $n$ is the number of observations in the data set.

STEP 4

Calculate the number of observations in the data set.
$n = 6$

STEP 5

Now, plug in the value of $n$ into the formula to find the position of Q3.
$\text{Position of Q3} = \frac{3}{4}(6 + 1)$

STEP 6

Calculate the position of Q3.
$\text{Position of Q3} = \frac{3}{4} \times 7 = \frac{21}{4} = 5.25$

STEP 7

Since the position of Q3 is not an integer, it falls between the 5th and 6th values in the ordered list. We will need to interpolate to find the exact value of Q3.

STEP 8

Identify the 5th and 6th values in the ordered list.
$\text{5th value} = 170$ $\text{6th value} = 190$

STEP 9

Since the position of Q3 is 5.25, it means Q3 is one-fourth of the way between the 5th and 6th values. We can calculate Q3 using linear interpolation:
$Q3 = \text{5th value} + \left(\text{Position of Q3} - \text{integer part of Position of Q3}\right) \times (\text{6th value} - \text{5th value})$

STEP 10

Calculate the fractional part of the position of Q3.
$\text{Fractional part of Position of Q3} = 5.25 - 5 = 0.25$

STEP 11

Now, plug in the values to calculate Q3.
$Q3 = 170 + (0.25 \times (190 - 170))$

STEP 12

Calculate the difference between the 6th and 5th values.
$190 - 170 = 20$

STEP 13

Multiply the fractional part by the difference calculated in STEP_12.
$0.25 \times 20 = 5$

STEP 14

Add the result from STEP_13 to the 5th value to find Q3.
$Q3 = 170 + 5$

STEP 15

Calculate the value of Q3.
$Q3 = 175$
The third quartile (Q3) for the given set of lengths is 175.

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Find the third quartile for the set of lengths: 130, 170, $160, 160, 150, 190$ .

STEP 1

Assumptions
1. We are given a set of lengths: 130, 170, 160, 160, 150, 190.
2. We need to find the third quartile (Q3) of this data set.
3. The third quartile is the value below which 75% of the data falls.
4. The data must be ordered from smallest to largest before calculating the quartiles.

STEP 2

First, we will order the given lengths from smallest to largest.
$\text{Ordered lengths: } 130, 150, 160, 160, 170, 190$

STEP 3

To find the third quartile, we need to determine the position in the ordered list where 75% of the data lies below it. The position can be found using the formula:
$\text{Position of Q3} = \frac{3}{4}(n + 1)$
where $n$ is the number of observations in the data set.

STEP 4

Calculate the number of observations in the data set.
$n = 6$

STEP 5

Now, plug in the value of $n$ into the formula to find the position of Q3.
$\text{Position of Q3} = \frac{3}{4}(6 + 1)$

STEP 6

Calculate the position of Q3.
$\text{Position of Q3} = \frac{3}{4} \times 7 = \frac{21}{4} = 5.25$

STEP 7

Since the position of Q3 is not an integer, it falls between the 5th and 6th values in the ordered list. We will need to interpolate to find the exact value of Q3.

STEP 8

Identify the 5th and 6th values in the ordered list.
$\text{5th value} = 170$ $\text{6th value} = 190$

STEP 9

Since the position of Q3 is 5.25, it means Q3 is one-fourth of the way between the 5th and 6th values. We can calculate Q3 using linear interpolation:
$Q3 = \text{5th value} + \left(\text{Position of Q3} - \text{integer part of Position of Q3}\right) \times (\text{6th value} - \text{5th value})$

STEP 10

Calculate the fractional part of the position of Q3.
$\text{Fractional part of Position of Q3} = 5.25 - 5 = 0.25$

STEP 11

Now, plug in the values to calculate Q3.
$Q3 = 170 + (0.25 \times (190 - 170))$

STEP 12

Calculate the difference between the 6th and 5th values.
$190 - 170 = 20$

STEP 13

Multiply the fractional part by the difference calculated in STEP_12.
$0.25 \times 20 = 5$

STEP 14

Add the result from STEP_13 to the 5th value to find Q3.
$Q3 = 170 + 5$

STEP 15

Calculate the value of Q3.
$Q3 = 175$
The third quartile (Q3) for the given set of lengths is 175.

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Find the third quartile for the set of lengths: 130, 170, 160,160,150,190160, 160, 150, 190160,160,150,190.

STEP 1

Assumptions1. We are given a set of lengths: 130, 170, 160, 160, 150, 190.2. We need to find the third quartile (Q3) of this data set.3. The third quartile is the value below which 75% of the data falls.4. The data must be ordered from smallest to largest before calculating the quartiles.

STEP 2

First, we will order the given lengths from smallest to largest.Ordered lengths: 130,150,160,160,170,190\text{Ordered lengths: } 130, 150, 160, 160, 170, 190Ordered lengths: 130,150,160,160,170,190

STEP 3

STEP 4

Calculate the number of observations in the data set.n=6n = 6n=6

STEP 5

Now, plug in the value of nnn into the formula to find the position of Q3.Position of Q3=34(6+1)\text{Position of Q3} = \frac{3}{4}(6 + 1)Position of Q3=43​(6+1)

STEP 6

Calculate the position of Q3.Position of Q3=34×7=214=5.25\text{Position of Q3} = \frac{3}{4} \times 7 = \frac{21}{4} = 5.25Position of Q3=43​×7=421​=5.25

STEP 7

Since the position of Q3 is not an integer, it falls between the 5th and 6th values in the ordered list. We will need to interpolate to find the exact value of Q3.

STEP 8

Identify the 5th and 6th values in the ordered list.5th value=170\text{5th value} = 1705th value=170 6th value=190\text{6th value} = 1906th value=190

STEP 9

STEP 10

Calculate the fractional part of the position of Q3.Fractional part of Position of Q3=5.25−5=0.25\text{Fractional part of Position of Q3} = 5.25 - 5 = 0.25Fractional part of Position of Q3=5.25−5=0.25

STEP 11

Now, plug in the values to calculate Q3.Q3=170+(0.25×(190−170))Q3 = 170 + (0.25 \times (190 - 170))Q3=170+(0.25×(190−170))

STEP 12

Calculate the difference between the 6th and 5th values.190−170=20190 - 170 = 20190−170=20

STEP 13

Multiply the fractional part by the difference calculated in STEP_12.0.25×20=50.25 \times 20 = 50.25×20=5

STEP 14

Add the result from STEP_13 to the 5th value to find Q3.Q3=170+5Q3 = 170 + 5Q3=170+5

STEP 15

Calculate the value of Q3.Q3=175Q3 = 175Q3=175The third quartile (Q3) for the given set of lengths is 175.

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Find the third quartile for the set of lengths: 130, 170, $160, 160, 150, 190$ .

Assumptions
1. We are given a set of lengths: 130, 170, 160, 160, 150, 190.
2. We need to find the third quartile (Q3) of this data set.
3. The third quartile is the value below which 75% of the data falls.
4. The data must be ordered from smallest to largest before calculating the quartiles.

First, we will order the given lengths from smallest to largest.
$\text{Ordered lengths: } 130, 150, 160, 160, 170, 190$

Calculate the number of observations in the data set.
$n = 6$

Now, plug in the value of $n$ into the formula to find the position of Q3.
$\text{Position of Q3} = \frac{3}{4}(6 + 1)$

Calculate the position of Q3.
$\text{Position of Q3} = \frac{3}{4} \times 7 = \frac{21}{4} = 5.25$

Identify the 5th and 6th values in the ordered list.
$\text{5th value} = 170$ $\text{6th value} = 190$

Calculate the fractional part of the position of Q3.
$\text{Fractional part of Position of Q3} = 5.25 - 5 = 0.25$

Now, plug in the values to calculate Q3.
$Q3 = 170 + (0.25 \times (190 - 170))$

Calculate the difference between the 6th and 5th values.
$190 - 170 = 20$

Multiply the fractional part by the difference calculated in STEP_12.
$0.25 \times 20 = 5$

Add the result from STEP_13 to the 5th value to find Q3.
$Q3 = 170 + 5$

Calculate the value of Q3.
$Q3 = 175$
The third quartile (Q3) for the given set of lengths is 175.