ACT mathematics score for a particular year are normally distributed with a mean of 27 and a standard deviation of 2 points.

Part A: What is the probability that a randomly selected score is greater than 29 points?

Part B: What percentage of students scores are between 31 and 23?

Part C: A student who scores 31 is in the _ percentile.

I have to solve using the empirical rule

Can somebody help me please? These are really complicated

A: 16%

B: 95%

C: 97.5%

Step-by-step explanation:

According to the empirical rule:

68% of a normal distribution is between -1 and +1 standard deviations.

95% of a normal distribution is between -2 and +2 standard deviations.

99.7% of a normal distribution is between -3 and +3 standard deviations.

Given μ = 27 and σ = 2.

Part A

29 is one standard deviation above the mean.  We can show this by calculating the z-score:

z = (x − μ) / σ

z = (29 − 27) / 2

z = 1

We know that 68% is between -1 and +1 standard deviations.  Since normal distributions are symmetrical, we can also say that 34% is between 0 and +1 standard deviations.

P(0 < Z < 1) = 68%/2 = 34%

We can also say that 50% is less than 0 standard deviations.

P(Z < 0) = 50%

Therefore, P(Z < 1) = 34% + 50% = 84%.

Which means P(Z > 1) = 100% − 84% = 16%.

Part B

Like before, calculate the z-scores:

z₁ = (23 − 27) / 2

z₁ = -2

z₂ = (31 − 27) / 2

z₂ = 2

From the empirical rule, we know this is 95% of the normal distribution.

Part C

We found in part B that the z-score is 2.

P(0 < Z < 2) = 95%/2 = 47.5%

P(Z < 2) = 50% + 47.5% = 97.5%

you are playing an adventure game on your computer. you come across a troll who won't let you pass and there is no other way to get to the other side of the river except over the bridge he is guarding.

the troll is a nerd. he takes great delight in watching people struggle with his problems. very few get them. he only gives you one chance and then he changes the puzzle. you have to solve this one.

before you are four blocks. they are numbered 11,6, -7,21.

he tells that the answer he wants is minus and between -1 and - 2. you can use +, -, ×, ÷   ( and ) as many times as you like or no times at all. you don't like the way he's looking at you, but you need to cross the bridge. what do you do?