Question

a silent auction was held for a quilt donated for charity. fifteen bids that were drawn randomly are listed here. find the value of the 45th percentile. $85 $240 $120 $60 $315 $190 $145 $105 $285 $260 $95 $215 $235 $175 $155 (1 point) $145 $150 $7 $155

Explanation:

Step1: Sort the data

Sort the bids in ascending order:
\(60,85,95,105,120,145,155,175,190,215,235,240,260,285,315\)

Step2: Calculate the position of the 45th percentile

Use the formula \(i = \frac{p}{100} \times n\), where \(p = 45\) and \(n=15\)
\(i=\frac{45}{100}\times15 = 6.75\)

Step3: Interpolate to find the value

Since \(i\) is not an integer, we use the formula \(x_i=x_{k}+(x_{k + 1}-x_{k})\times f\), where \(k = 6\) (the integer part of \(6.75\)), \(f=0.75\) (the decimal part of \(6.75\)), \(x_{6}=155\), \(x_{7}=175\)
\(x_{6.75}=155+(175 - 155)\times0.75\)
\(=155+20\times0.75\)
\(=155 + 15\)
\(=170\)
Wait, there is a mistake above. Let's use another formula. For percentile, when \(n = 15\) and \(p=45\), \(i=(n + 1)\times\frac{p}{100}=(15 + 1)\times\frac{45}{100}=16\times0.45 = 7.2\)
\(x_{7.2}=x_7+(x_8 - x_7)\times0.2\)
\(x_7 = 155\), \(x_8=175\)
\(x_{7.2}=155+(175 - 155)\times0.2\)
\(=155+20\times0.2\)
\(=155 + 4\)
\(=159\)
Another correct way:
Sort the data: \(60,85,95,105,120,145,155,175,190,215,235,240,260,285,315\)
\(n = 15\), \(p = 45\)
\(L=(n + 1)\times\frac{p}{100}=(15+1)\times\frac{45}{100}=7.2\)
The 7th value is \(155\) and the 8th value is \(175\)
\(P_{45}=155+(175 - 155)\times0.2=155 + 4=159\)
But if we use \(i=\frac{p}{100}\times n=\frac{45}{100}\times15 = 6.75\)
\(x_{6.75}=x_6+(x_7 - x_6)\times0.75\)
\(x_6 = 145\), \(x_7=155\)
\(x_{6.75}=145+(155 - 145)\times0.75=145 + 7.5=152.5\)
Wait, no. The standard formula for percentile:
If \(n\) is the number of data points.
\(index=i=\frac{p}{100}\times n\)
When \(i\) is not an integer, round up to the next integer.
\(i=\frac{45}{100}\times15 = 6.75\), round up to \(7\)
The 7th value in the sorted data (\(60,85,95,105,120,145,155,175,190,215,235,240,260,285,315\)) is \(155\)

Answer:

\(155\) (the fourth option)