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# 4) The correlation coefficient r is a sample statistic. What does it tell us about the...

4) The correlation coefficient r is a sample statistic. What does it tell us about the value of the population correlation coefficient ρ (Greek letter rho)? You do not know how to build the formal structure of hypothesis tests of ρ yet. However, there is a quick way to determine if the sample evidence based on ρ is strong enough to conclude that there is some population correlation between the variables. In other words, we can use the value of r to determine if ρ ≠ 0. We do this by comparing the value |r| to an entry in the correlation table. The value of α in the table gives us the probability of concluding that ρ ≠ 0 when, in fact, ρ = 0 and there is no population correlation. We have two choices for α: α = 0.05 or α = 0.01. 10-table-06.gif

(a) Look at the data below regarding the variables x = age of a Shetland pony and y = weight of that pony. Is the value of |r| large enough to conclude that weight and age of Shetland ponies are correlated? Use α = 0.05. (Use 3 decimal places.) x 3 6 12 20 21 y 60 95 140 170 174 r= critical r=

(b) Look at the data below regarding the variables x = lowest barometric pressure as a cyclone approaches and y = maximum wind speed of the cyclone. Is the value of |r| large enough to conclude that lowest barometric pressure and wind speed of a cyclone are correlated? Use α = 0.01. (Use 3 decimal places.) x 1004 975 992 935 980 938 y 40 100 65 145 66 148 r= critical r=

5) Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and ybe random variables with means μx and μy, variances σ2x and σ2y, and population correlation coefficient ρ(the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. μw = aμx + bμy σ2w = a2σ2x + b2σ2y + 2abσxσyρ In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let x represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let y represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates. μx ≈ 7.35, σx ≈ 6.58, μy ≈ 13.17, σy ≈ 18.59, ρ ≈ 0.421

(b) Suppose you decide to put 65% of your investment in bonds and 35% in real estate. This means you will use a weighted average w = 0.65x + 0.35y. Estimate your expected percentage return μw and risk σw. μw = σw = (c) Repeat part (b) if w = 0.35x + 0.65y. μw = σw =

6) In the least-squares line ŷ = 5 + 7x, what is the marginal change in ŷ for each unit change in x?

7) The following Minitab display gives information regarding the relationship between the body weight of a child (in kilograms) and the metabolic rate of the child (in 100 kcal/ 24 hr). Predictor Coef SE Coef T P Constant 0.8543 0.4148 2.06 0.84 Weight 0.41969 0.02978 13.52 0.000 S = 0.517508 R-Sq = 95.2%

(a) Write out the least-squares equation. y hat = + x

(b) For each 1 kilogram increase in weight, how much does the metabolic rate of a child increase? (Use 5 decimal places.)

(c) What is the value of the correlation coefficient r? (Use 3 decimal places.)

(6) Ans: 7

(7)

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