Question

4. Summary Statistics (a) Obtain for the following cases with n 5: (1)1. (2). (3) 22. (4) (b) Compute from first principles (i.e. using the formula and a calculator) the mean, standard deviation, coefficient of variation and symmetry statistic for the sample 10, 6. 1, 7. Show all calculations (c) IQ scores have a mean of 100, standard deviation of 14 and are approximately normally distributed. What range of IQ scores do you expect 67% of the population to lie in?

Answer 1

c v = alpha/mu = 50(x)/mean (y) = squareroot 14/6 symmetric statistic = skewnes statistic skewness = gamma_1 = E[(x: - 4/alpha)^2] = -0.7613 therefor this sample is left skewed because gamma_1 < 0 (c) mu_1 v = 100 alpha_1 v = 14 for normal distribution 67 % is constituted in an interact (mu - 6, mu + 6) therefore (86, 114) is the interval in which 67 % of population exists \r\n (a) sigma^s_i = 1 z_i: z_1 = 1 - z_1 = ..... z_5 = 1 sigma z_i = 1 + 1 + 1 + 1 + 1 = 5 z_i = i z_i = 2 i^2 sigma z_i = sigma i = n(n + 1)/2 = 5(6)/2 = 15 sigma z_i = 2 sigma i^2 = n(n + 1) (2 n + 1)/6 middot 2 = n(n + 1) (2 n + 1)/3 = 5(6) (11)/3 = 110 z_i = 1/i sigma z_i = 1 + 1/2 + 1/3 + 1/4 + 1/5 = 60 + 30 + 20 + 15 + 12/60 = 137/60 = 2.29 (b) sample (10, 6, 1, 7) mean = 10 + 6 + 1 + 7/4 = 6 var sigma (x_i - x)^2/n - 1 = (10 - 6