The length of time taken on the SAT for a group of students is normally distributed with a mean of 2.5 hours and a standard deviation of 0.25 hours. A sample size of n = 60 is drawn randomly from the population. Find the probability that the sample mean is between two hours and three hours.

Question

Mathematics

Monserrat Livingston

10 April, 09:17

The length of time taken on the SAT for a group of students is normally distributed with a mean of 2.5 hours and a standard deviation of 0.25 hours. A sample size of n = 60 is drawn randomly from the population. Find the probability that the sample mean is between two hours and three hours.

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Ponce · Answer 1

Step-by-step explanation:

Since the length of time taken on the SAT for a group of students is normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - u) / s

Where

x = length of time

u = mean time

s = standard deviation

From the information given,

u = 2.5 hours

s = 0.25 hours

We want to find the probability that the sample mean is between two hours and three hours ... It is expressed as

P (2 lesser than or equal to x lesser than or equal to 3)

For x = 2,

z = (2 - 2.5) / 0.25 = - 2

Looking at the normal distribution table, the probability corresponding to the z score is 0.02275

For x = 3,

z = (3 - 2.5) / 0.25 = 2

Looking at the normal distribution table, the probability corresponding to the z score is 0.97725

P (2 lesser than or equal to x lesser than or equal to 3)

= 0.97725 - 0.02275 = 0.9545