Question

Standard Score

All problems below with the heading "Standard Score" are based on the data in this problem.

A researcher at Nintendo has access to millions of smartphones. He finds that the mean number of monsters captured each day by people with the Pokémon Go app is 286. The standard deviation for number of monsters captured is per day is 40. The number of monsters captured per day is normally distributed. (It is OK if you don't understand the game. What we have is a normally distributed variable with a mean and standard deviation.)

What percentage of users on a given day capture more than 300 Pokemons?

What percentage of users on a given day capture between 250 and 350 Pokemons?

What number of pokemons were captured at the 25th percentile?

Answer 1

Solution :

Given that ,

mean = = 286

standard deviation = = 40

P(x > 300) = 1 - P(x < 300)

= 1 - P[(x - ) / < (300 - 286) / 40]

= 1 - P(z < 0.35)

= 1 - 0.6368

= 0.3632

Percentage = 36.32%

P(250 < x < 350) = P[(250 - 286)/ 40) < (x - ) /  < (350 - 286) / 40) ]

= P(-0.9 < z < 1.6)

= P(z < 1.6) - P(z < -0.9)

= 0.9452 - 0.1841

= 0.7611

Percentage = 76.11%

Using standard normal table,

P(Z < z) = 25%

P(Z < -0.67) = 0.25

z = -0.67

Using z-score formula,

x = z * +

x = -0.67 * 40 + 286 = 259.2

25th percentile = 259.2