Question

In Hoop Fever, a player has 60 seconds to make as many baskets as possible. Morgan and Tim play head-to-head every Tuesday. Let M = the number of baskets made by Morgan and T= the number of baskets made by Tim in a randomly selected match. Based on previous matches, we know that ow = 5.7 and or = 10.3. Assume that these two random variables are independent. Define D= M - T. Earlier, we found that up = 8.6. Calculate and interpret the standard deviation of D.

Answer 1

The standard deviation of the difference in the number of baskets made by Morgan and Tim is approximately 11.77, representing the variability of Morgan's lead over Tim from match to match.

Step-by-step explanation:

To calculate the standard deviation of the difference between the number of baskets made by Morgan (M) and Tim (T), denoted as D, you use the formula for the standard deviation of the difference between two independent random variables. Since M and T are independent, the variance of D is the sum of the variances of M and T. We are given the standard deviations for M and T, which are 5.7 and 10.3 respectively.

To find the variance of D, we square the standard deviations and add them together:

Var(M) = σ²M = 5.7²

Var(T) = σ²T = 10.3²

Var(D) = Var(M) + Var(T) = 5.7² + 10.3²

σD = √(Var(D)) = √(σ²M + σ²T)

Performing the calculation, we have:

σD = √(5.7² + 10.3²) = √(32.49 + 106.09) = √(138.58) ≈ 11.77

The standard deviation of D, which is the difference in the number of baskets made by Morgan and Tim, is approximately 11.77. This measures how much the difference in their scores varies from one match to another. Knowing that the mean difference (M - T) is 8.6, we can interpret the standard deviation as indicating how spread out Morgan's lead over Tim is likely to be from game to game.

Answer 2

To calculate the standard deviation of D, find the variance of D by summing the variances of M and T. Then, take the square root of the variance to find the standard deviation.

Step-by-step explanation:

To calculate the standard deviation of D, we first need to find the variance of D. The variance of D is given by the sum of the variances of M and T, since M and T are independent random variables. So, Var(D) = Var(M) + Var(T). The standard deviation of D is then the square root of the variance of D.

Given that up = 8.6, we can write the equation Var(D) = Var(M) + Var(T) = Var(M) + Var(-M) = 2Var(M), where Var(-M) represents the variance of -M. Since Var(-M) = Var(M), we can simplify the equation as Var(D) = 2Var(M).

Using the formula for variance, Var(D) = 2 * (ow)² = 2 * (5.7)² = 64.98.

The standard deviation of D is then the square root of the variance, which is √64.98 = 8.06.