q14. using the data values given, answer parts a through e.\n5 5 5 5 12 6 2 2 2 2 7 9 11 8 7\na. find the…

Part a: Find the mean

Explanation:

Step1: Count the number of data points

There are 15 data points: (5, 5, 5, 5, 12, 6, 2, 2, 2, 2, 7, 9, 11, 8, 7)

Step2: Calculate the sum of the data

[ \begin{align*} \text{Sum} &= 5\times4 + 12 + 6 + 2\times4 + 7\times2 + 9 + 11 + 8\ &= 20 + 12 + 6 + 8 + 14 + 9 + 11 + 8\ &= 20+12=32; 32+6=38; 38+8=46; 46+14=60; 60+9=69; 69+11=80; 80+8=88 \end{align*} ]

Step3: Calculate the mean (Mean = Sum / Number of data points)

[ \text{Mean} = \frac{88}{15} \approx 5.87 ]

Answer:

(\approx 5.87)

Part b: Find the median

Explanation:

Step1: Arrange the data in ascending order

(2, 2, 2, 2, 5, 5, 5, 5, 6, 7, 7, 8, 9, 11, 12)

Step2: Find the middle value (since there are 15 data points, the 8th value is the median)

The 8th value is (5)

Answer:

(5)

Part c: Find the mode

Explanation:

Step1: Identify the value(s) with the highest frequency

• (2) appears 4 times.
• (5) appears 4 times.
• Other values have lower frequencies.

Step2: Determine the mode(s)

Both (2) and (5) are modes as they appear most frequently (4 times each).

Answer:

(2) and (5)

Part d: Find the standard deviation

Explanation:

Step1: Recall the formula for sample standard deviation: (s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}) (where (\bar{x}) is the mean, (x_i) are data points, (n) is the number of data points)

We know (\bar{x}=\frac{88}{15}\approx5.87), (n = 15)

Step2: Calculate ((x_i - \bar{x})^2) for each data point and sum them up

[ \begin{align*} &\text{For } x = 2: (2 - \frac{88}{15})^2 = (\frac{30 - 88}{15})^2 = (\frac{-58}{15})^2=\frac{3364}{225}\text{ (4 times)}\ &\text{For } x = 5: (5 - \frac{88}{15})^2 = (\frac{75 - 88}{15})^2 = (\frac{-13}{15})^2=\frac{169}{225}\text{ (4 times)}\ &\text{For } x = 6: (6 - \frac{88}{15})^2 = (\frac{90 - 88}{15})^2 = (\frac{2}{15})^2=\frac{4}{225}\ &\text{For } x = 7: (7 - \frac{88}{15})^2 = (\frac{105 - 88}{15})^2 = (\frac{17}{15})^2=\frac{289}{225}\text{ (2 times)}\ &\text{For } x = 8: (8 - \frac{88}{15})^2 = (\frac{120 - 88}{15})^2 = (\frac{32}{15})^2=\frac{1024}{225}\ &\text{For } x = 9: (9 - \frac{88}{15})^2 = (\frac{135 - 88}{15})^2 = (\frac{47}{15})^2=\frac{2209}{225}\ &\text{For } x = 11: (11 - \frac{88}{15})^2 = (\frac{165 - 88}{15})^2 = (\frac{77}{15})^2=\frac{5929}{225}\ &\text{For } x = 12: (12 - \frac{88}{15})^2 = (\frac{180 - 88}{15})^2 = (\frac{92}{15})^2=\frac{8464}{225}\ \end{align*} ]

Step3: Calculate the sum of squared deviations

[ \begin{align*} \text{Sum of squared deviations}&= 4\times\frac{3364}{225}+ 4\times\frac{169}{225}+\frac{4}{225}+ 2\times\frac{289}{225}+\frac{1024}{225}+\frac{2209}{225}+\frac{5929}{225}+\frac{8464}{225}\ &=\frac{13456 + 676 + 4 + 578 + 1024 + 2209 + 5929 + 8464}{225}\ &=\frac{13456+676=14132; 14132+4=14136; 14136+578=14714; 14714+1024=15738; 15738+2209=17947; 17947+5929=23876; 23876+8464=32340}{225}\ &=\frac{32340}{225}= 143.733\cdots \end{align*} ]

Step4: Calculate the variance (Variance = Sum of squared deviations / (n - 1))

[ \text{Variance} = \frac{32340/225}{14}=\frac{32340}{225\times14}=\frac{32340}{3150}\approx10.27 ]

Step5: Calculate the standard deviation (Standard Deviation = (\sqrt{\text{Variance}}))

[ \text{Standard Deviation} = \sqrt{10.27} \approx 3.20 ]

Answer:

(\approx 3.20)

Part e: Determine the percentile rank of 2

Explanation:

Step1: Recall the formula for percentile rank: (\text{Percentile Rank} = \frac{\text{Number of values less than } x + 0.5\times\text{Number of values equal to } x}{n}\times100)

Here, (x = 2), (n = 15)

Step2: Count the number of values less than 2 (0) and equal to 2 (4)

Step3: Calculate the percentile rank

[ \text{Percentile Rank} = \frac{0 + 0.5\times4}{15}\times100=\frac{2}{15}\times100\approx13.33 ]

Answer:

(\approx 13.33)