uppose IQ scores were obtained for 20 randomly selected sets of couples. The 20 pairs of measurements yield \( \bar{x}=95.68, \bar{y}=93.05, r=0.863, P \)-value \( =0.000 \), and \( =10.14+0.87 x \), where \( x \) represents the IQ score of the wife. Find the best predicted value of \( \hat{y} \) given that the wife has an IQ of 103 ? Use a significance level of 0.05 . Click the icon to view the critical values of the Pearson correlation coefficient \( r \).

To find the best predicted value of \( \hat{y} \) given that the wife has an IQ of 103, we'll use the linear regression equation provided, which is \( \hat{y} = 10.14 + 0.87x \). Substituting the wife's IQ score \( x = 103 \) into the equation: \[ \hat{y} = 10.14 + 0.87(103) \] Calculating this gives: \[ \hat{y} = 10.14 + 89.61 = 99.75 \] So, the best predicted value of \( \hat{y} \) when the wife has an IQ of 103 is approximately 99.75. Since the \( P \)-value is much lower than the significance level of 0.05, we can confidently say that there is a significant linear relationship between the IQ scores of husbands and wives. Now, let's move on to additional fun and engaging insights! Did you know that intelligence testing has a rich history? It began in the early 20th century, with pioneers like Alfred Binet and Lewis Terman creating standardized tests to identify students needing extra support. IQ tests have since evolved, sparking debates about the nature of intelligence, cultural bias, and their implications in education and psychology. Furthermore, the application of IQ tests extends beyond academia. They’ve been influential in various fields such as hiring practices, military placement, and even assessing cognitive abilities in aging populations. Knowing how decisions are influenced by these scores can shed light on their societal impact and the importance of understanding their context and limitations!