Suppose IQ scores were obtained for 20 randomly selected sets of siblings. The 20 pairs of measurements yield ( ar{x}=98.24, ar{y}=98, r=0.881 ), ( P )-value ( =0.000 ), and ( hat{y}=-4.83+1.05 x ), where ( x ) represents the IQ score of the older child. Find the best predicted value of ( hat{y} ) given that the older child has an IQ of 90 ? Use a significance level of 0.05 . Click the icon to view the critical values of the Pearson correlation coefficient ( r ). The best predicted value of ( hat{y} ) is (Round to two decimal places as needed.) Critical Values of the Pearson Correlation Coefficient ( r ) Critical Values of the Pearson Correlation Coefficient ( r ) egin{tabular}{|l|l|l|} hline ( mathbf{n} ) & ( oldsymbol{alpha}=mathbf{0 . 0 5} ) & ( oldsymbol{alpha}=mathbf{0 . 0 1} ) \ hline 4 & 0.950 & 0.990 \ hline 5 & 0.878 & 0.959 \ hline 6 & 0.811 & 0.917 \ hline 7 & 0.754 & 0.875 \ hline 8 & 0.707 & 0.834 \ hline 9 & 0.666 & 0.798 \ hline 10 & 0.632 & 0.765 \ hline 11 & 0.602 & 0.735 \ hline 12 & 0.576 & 0.708 \ hline 13 & 0.553 & 0.684 \ hline 14 & 0.532 & 0.661 \ hline 15 & 0.514 & 0.641 \ hline 16 & 0.497 & 0.623 \ hline 17 & 0.482 & 0.606 \ hline 18 & 0.468 & 0.590 \ hline 19 & 0.456 & 0.575 \ hline 20 & 0.444 & 0.561 \ hline 25 & 0.396 & 0.505 \ hline 30 & 0.361 & 0.463 \ hline 35 & 0.335 & 0.430 \ hline 40 & 0.312 & 0.402 \ hline 45 & 0.294 & 0.378 \ hline 50 & 0.279 & 0.361 \ hline 60 & 0.254 & 0.330 \ hline 70 & 0.236 & 0.305 \ hline 80 & 0.220 & 0.286 \ hline 90 & 0.207 & 0.269 \ hline 100 & 0.196 & 0.256 \ hline ( mathbf{n} ) & ( oldsymbol{alpha}=mathbf{0 . 0 5} ) & ( oldsymbol{alpha}=mathbf{0 . 0 1} ) \ hline & & \ hline end{tabular}
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There are 10 students in class and the number of siblings in their families are given by {1,1,1,1,2,3,3,4,4,4}. Consider the experiment of randomly choosing one of these students and letting Y denote the number of siblings in this chosen student's family: (a) Find the probability mass function (pmf) of Y and plot this pmf. (b) Find Pr{Y > 2}. (c) Find the mean, μ, of Y. (d) Find the standard deviation, σ, of Y. (e) Find the exact value of P(μ - 2σ ≤ Y ≤ μ + 2σ). Still considering the setting in Problem 1, suppose now that the random experiment is to randomly choose two students from the 10 students, without replacement, and then observing the number of siblings in their families. (a) Draw a tree diagram showing the possible outcomes of this experiment, together with their probabilities. (b) What is the probability of the outcome (1, 1)? How about the outcome (2, 3)? (c) Define the random variable X to be the average of the values in your sample. Thus, if the outcome is (1, 1), then X = (1+1)/2 = 1; while if the outcome is (2, 3), then X = (2+3)/2 = 2.5. Find the probability mass function (pmf) of X, then plot this pmf. (d) Compare the pmf of X with the pmf of Y you obtained in Problem 1. (e) Compute the mean of X. Is this equal to the mean of Y in Problem 1? (f) Compute the standard deviation of X. How does this compare with the standard deviation of Y in Problem 1.
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Fifty children from each of two neighborhoods are given identical reading tests. The researcher is primarily interested in whether children in the two neighborhoods score significantly differently from each other. The researcher is also interested in how a child's age plays into the relationship between neighborhood and reading score. A regression was run with the variables defined as: Y, reading score; Xage, age in years and months; Xnbd, which is zero for children from neighborhood A and one for children from neighborhood B. The following are the standard regression output and the sequential Sum of Squares table for the regression: The model was: Y = β0 + β1 Xage + β2 Xnbd + β3 Xage:Xnbd + ε, where ε ~ N(0, σ^2). Coefficients: (Intercept) 16.457 12.985 1.267 0.208097 Xage 5.046 1.481 3.406 0.000964 Xnbd -36.304 18.812 -1.930 0.056584 Xage:Xnbd 4.521 2.192 2.062 0.041863 Residual standard error: 22.09 on 96 degrees of freedom Multiple R-squared: 0.3273, Adjusted R-squared: 0.3062 F-statistic: 15.57 on 3 and 96 DF, p-value: 2.508e-08 Sequential Sum of Squares Table (Response: y): Xage 1 20669 Xnbd 1 49 Xage:Xnbd 1 2076 Residuals 96 46858
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