Question

(1 point) Suppose a certain score in a game may be modeled as follows: We will spin the wheel below 100 times and sum the numbers. The expected value of the sum of 100 spins is points with a standard error of points. The probability that the sum of 100 spins will be between 245 and 255, inclusively, is approximately , using a Normal approximation with a continuity correction. (Answer to four decimal places.)

Answer 1

Answer:

a) Expected sum of numbers obtained with 100 spins = 260

b) Standard deviation of the sum of numbers obtained with 100 spins = 12

c) Probability that the sum of numbers obtained on 100 spins is 245 to 255, inclusive, using the correction factor = 0.25344

Step-by-step explanation:

Before solving for expected value and standard deviation of a 100 spins, we do for one spin first,

Expected value is given as

E(X) = Σ xᵢpᵢ

where xᵢ = each variable

pᵢ = probability of each variable

We have sample spaces 1, 2, and 4

with 1 appearing once, 2 appearing twice and 4 appearing twice too. So, their respective probabilities of turning up are (1/5), (2/5) and (2/5).

E(X) = (1)(1/5) + (2)(2/5) + (4)(2/5) = 2.6

Standard deviation = √(variance)

Variance = Var(X) = Σx²p − μ²

where μ = E(X) = 2.6

Σx²p = (1²)(1/5) + (2²)(2/5) + (4²)(2/5) = 8.2

Variance = 8.2 - 2.6² = 1.44

Standard deviation = √(variance) = √(1.44) = 1.2

Now for 100 spins,

E(100X) = 100 × E(X) = 100 × 2.6 = 260

Standard deviation (100X) = √100 × 1.2 = 10 × 1.2 = 12

Probability that the sum of numbers obtained on 100 spins is 245 to 255, inclusive, using the correction factor.

P(245 ≤ x ≤ 255)

Introducing the correction factor, we have

P(244.5 ≤ x ≤ 255.5) (correction factor is 0.5)

Using normal distribution tables, we first standardize 244.5 and 255.5.

The standardized score for any value is the value minus the mean then divided by the standard deviation.

Note that the mean = μ = 260

Standard deviation = σ = 12

For 244.5

z = (x - μ)/σ = (244.5 - 260)/12 = - 1.29

For 255.5

z = (x - μ)/σ = (255.5 - 260)/12 = - 0.38

To determine the required probability

P(244.5 ≤ x ≤ 255.5) = P(-1.29 ≤ z ≤ -0.38)

We'll use data from the normal probability table for these probabilities

P(244.5 ≤ x ≤ 255.5) = P(-1.29 ≤ z ≤ -0.38)

= P(z ≤ -0.38) - P(z ≤ -1.29)

= 0.35197 - 0.09853

= 0.25344

Hope this Helps!!!