Question

The attendance at baseball games at a certain stadium is normally distributed, with a mean of 30,000 and a standard deviation of 1500. For any given game: A) What is the probability that attendance is greater than 27,300? B) What is the probability that attendance will be 30,000 or more? C) What is the probability of attendance between 27,000 and 32,000? D) What must the attendance be at the game, for that game's attendance to be in the top 5% of all games? E) What is the probability that attendance is less than 33,000?

Answer 1

a) 0.964

b) 0.500

c) 0.885

d)
`x \geq 32467.5`

e) 0.997

Explanation:

We are given the following information in the question:

Mean, μ = 30000

Standard Deviation, σ = 1500

We are given that the distribution of attendance at stadium is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

a) P(attendance is greater than 27,300)

P(x > 27300)

`P( x > 27300) = P( z > \displaystyle(27300 - 30000)/(1500)) = P(z > -1.8)`

`= 1 - P(z \leq -1.8)`

Calculation the value from standard normal z table, we have,

`P(x > 27300) = 1 - 0.036 = 0.964 = 96.4\%`

b) P(attendance greater than or equal to 30000)

`P(x > 30000) = P(z > \displaystyle(30000-30000)/(1500)) = P(z \geq 0)\\\\P( z \geq 0) = 1 - P(z \leq 0)`

Calculating the value from the standard normal table we have,

`1 - 0.500 = 0.500 = 50\%`

c) P(attendance between 27000 and 32000)

`P(27000 \leq x \leq 32000) = P(\displaystyle(27000 - 30000)/(1500) \leq z \leq \displaystyle(32000-30000)/(1500)) = P(-2 \leq z \leq 1.33)\\\\= P(z \leq 1.33) - P(z < -2)\\= 0.908 - 0.023 = 0.885 = 88.5\%`

`P(27000 \leq x \leq 32000) = 88.5\%`

e) P(attendance less than 33000)

P(x < 33000)

`P( x < 33000) = P( z < \displaystyle(33000 - 30000)/(1500)) = P(z < 2)`

Calculating the value from the standard normal table we have,

P(x < 33000) = 0.997 = 99.7%

d) We have to find x such that:

`P( X > x) = P( z > \displaystyle(x - 30000)/(1500)) = 0.95`

Calculating the value from the standard normal table we have,

P(z = 1.645) = 0.95

Thus,

`\displaystyle(x - 30000)/(1500) \geq 1.645\\\\x \geq 32467.5`

The attendance should be greater than or equal to 32467.5 to be in the top 5% of all games.