Topic 4.9: Normal Distribution Questions

Explain why, in AI SL, you do NOT need to convert X to a z-score to find normal probabilities.

X ~ N(200, 25²). Find P(180 < X < 220).

X ~ N(40, 6²). Write down P(X < 40).

The contents of cans are N(330, 4²) ml. Find the probability a can has less than 325 ml.

For a normal distribution, state the relationship between the mean, median and mode.

State the approximate percentage of data within 1 standard deviation of the mean of a normal distribution.

A normal curve has mean 20. State what happens to the curve if the mean increases to 30 (σ unchanged).

X ~ N(50, 10²). Find P(X > 65).

Delivery times are N(30, 5²) minutes. Find the probability a delivery takes between 25 and 35 minutes.

Heights are N(170, 9²) cm. Find the probability a randomly chosen person is taller than 185 cm.

X ~ N(50, 10²). Find P(X < 60) using a GDC.

State the GDC command to find P(X < a) for X ~ N(μ, σ²).

The masses of apples are normally distributed with mean 150 g and standard deviation 20 g.

X ~ N(70, 8²). Find the value exceeded by the top 15% (i.e. find a with P(X > a) = 0.15).

Two factories produce bolts. Factory A's lengths are N(50, 2²) mm and Factory B's lengths are N(50, 5²) mm.

Heights are N(165, 8²) cm. Find the percentage of people between 149 cm and 181 cm.

X ~ N(25, 4²). Find the lower quartile (the value below which 25% of data lies).

The reaction times of drivers in a study are normally distributed with a mean of 0.30 seconds and a standard deviation of 0.04 seconds. Let T be the reaction time of a randomly chosen driver.

(a) Find the percentage of drivers with a reaction time between 0.26 and 0.34 seconds.

(b) Find the percentage of drivers with a reaction time less than 0.22 seconds.

A machine fills bottles with juice. The volume in each bottle is normally distributed with a mean of 500 ml and a standard deviation of 5 ml. Let X be the volume in a randomly chosen bottle.

(a) Find the percentage of bottles with a volume between 485 ml and 515 ml.

(b) Find the percentage of bottles with a volume greater than 510 ml.

X ~ N(500, 50²). Out of 1000 values, find the expected number greater than 600.