statistics homework 4

Statistics Homework 4 50 points Sum of Squared Errors (SS) = 2 𝑖=1 𝑛 ∑ (𝑥 𝑖 − µ) ● Step 1: Subtract each number from the mean. ● Step 2: Square each answer. ● Step 3: Add for final answer. Variance ( ) = σ 2 𝑆𝑆 𝑁 – Known as Standard Deviation σ = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 1. What does the variance measure? What are the symbols for population variance and sample variance? (Do not google the symbols, actually find them in your Word program. Insert-> Symbols -> More symbols -> Subset -> Greek & Coptic) (4 points) Variance measures how far each number in the set is from the mean, and thus from every number in the set. There are two types: the population variance, usually denoted by σ2 and the sample variance usually denoted by s2. 2. Answer the questions about standard deviation. (4 points) a. What does a large standard deviation tell us about our data? There is a lot of variance in the observed data around the mean. b. What does a small standard deviation tell us about our data? The data is more concentrated around the mean. 3. Calculate the mean, median, and mode for the following set of class grades: (4 points) 102, 93, 74, 52, 69, 71, 82, 77, 91 Mean - 102, 93, 74, 52, 69, 71, 82, 77, 91= 711/ 9= 79 Median- 52, 69, 71, 74, 77, 82, 91, 93, 102= 77

Mode- 0 4. Calculate the sum of squares, variance, and standard deviation of the following population of data: (7 points) 55, 115, 24, 80, 35, 15, 14, 12, 20, 40 Mean: 41 X X - µ ( X - 2 µ) 55 14 196 115 74 5,479 24 -17 289 80 39 1,521 35 -6 36 15 -26 676 14 -27 729 12 -29 841 20 -21 441 40 -1 1 Sum of squared errors (SS) = 2 𝑖=1 𝑛 ∑ (𝑥 𝑖 − µ) 10, 206 Variance ( ) = σ 2 𝑆𝑆 𝑁 1, 020.6 σ = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 31. 95 5. What is the mean for question 4? Would you say this fits the population data well? Why or why not? (4 points) The mean is 41. I would say that this does fit the population data well because it captures all the numbers.

213 123.8 15326.44 72 -17.2 295.84 9 -80.2 6432.04 150 60.8 3696.64 Sum of squared errors (SS) = 2 𝑖=1 𝑛 ∑ (𝑥 𝑖 − µ) 33354.8 Variance ( ) = σ 2 𝑆𝑆 𝑁 33,354.8 σ = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 373.93 The numbers changed because of the value of the mean which changed up all the numbers. 8. To understand how people felt about Taylor Swift’s new album, I asked my friends for their opinion, asking them to rank it from 1-10 (10 being the best they have ever heard). Using their scores, calculate the mean, standard deviation, median, and range for these responses. Do the mean and standard deviation appear to fit the data well? (7 points) 10, 2, 5, 9, 1, 6, 8, 9, 6, 4, 9, 7, 7, 6, 9, 6

Mean: 81 X X - µ ( X - 2 µ) 10 -71 5041 2 -79 6241 5 -76 5776 9 -72 5184 1 -80 6400 6 -75 5625 8 -73 5329 9 -72 5184 6 -75 5625 4 -77 5929 9 -72 5184 7 -74 5479 7 -74 5476 6 -75 5625 9 -72 5184 6 -75 5625 Sum of squared errors (SS) = 2 𝑖=1 𝑛 ∑ (𝑥 𝑖 − µ) 78,101 Variance ( ) = σ 2 𝑆𝑆 𝑁 7,810.1 σ = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 96.42 9. Using the data from question 8, construct a histogram (using Word or Excel) including one column for each number, a chart title, number of respondents as a subtitle, and an x-axis label. Does the data appear to have issues with kurtosis or skewness? (There are