Question

Gloria enjoys playing Skee ball an arcade game where players toss a ball so it rolls up a ramp and falls into one of several slots. Each slot is worth a different number of points based on how difficult it is to land the ball in that slot. Gloria only aims at the most difficult slot, which is worth 100 points, but she only has a 10% chance of landing the ball in that slot. If she misses, her ball will certainly land in the 10 point slot.

The table below displays the probability distribution of X, the number of points Gloria scores on a random shot.

X = points

100

10

P(X)

10%

90%

Given that μx = 19 points, calculate σχ.

Answer 1

Calculating the variance, we find that the standard deviation of Gloria's score on a random shot in Skee ball is 27 points.

Step-by-step explanation:

To calculate the standard deviation (often symbolized as σx or simply σ) of Gloria's score in Skee ball, we can use the formula for the standard deviation of a discrete random variable, which is:

σ = √[Σ(P(X) × (X - μx)^2)]

Given that μx (the mean of the random variable X) equals 19 points, we can plug in the values from the probability distribution:

• The probability of scoring 100 points (P(X=100)) is 10% or 0.10.
• The probability of scoring 10 points (P(X=10)) is 90% or 0.90.

We calculate the variance (σ2) first:

σ2 = [0.10 × (100 - 19)^2] + [0.90 × (10 - 19)^2]

σ2 = [0.10 × 81^2] + [0.90 × (-9)^2]

σ2 = [0.10 × 6561] + [0.90 × 81]

σ2 = 656.10 + 72.90

σ2 = 729

Now, we find the standard deviation by taking the square root of the variance:

σ = √729

σ = 27

Therefore, the standard deviation (σ) of Gloria's score on a random shot in Skee ball is 27 points.