The amount of time to complete a physical activity in a PE class is normally distributed with a mean of 35 seconds and a standard deviation of 3 seconds.

Question

a) What is the probability that a randomly chosen student completes the activity in less than 27 seconds?

b)What is the probability that a randomly chosen student completes the activity in more than 39 seconds?

c) What proportion of students take between 30 and 38 seconds to complete the activity?

d) 70% of all students finish in less than ? seconds.

e) What time in seconds is the cutoff for the fastest 10%?

f) Is completing the physical activity in less than 24 seconds unusual?

Answer 1

d) You need to first find the z-score for a proportion of 0.7. Based on the unit normal table, you want proportion of 0.7 in the body. This corresponds to a z-score of 0.53. Then, you use the z-score formula of z = (X-µ)/ σ. We plug in z=+0.53, µ = 35, and σ = 3.

The formula becomes: 0.53 = (X-35)/3. We solve for X, and X = 0.53(3)+35 = 36.59 seconds.

e) For this question, you can either look for a proportion of 0.10 in the tail or 0.90 in the body to get the z-score. z = -1.28. Then, you use the z-score formula of z = (X-µ)/ σ. We plug in z=-1.28, µ = 35, and σ = 3. The z is negative in this instance because faster times are smaller numbers.

The formula becomes: -1.28 = (X-35)/3. We solve for X, and X = -1.28(3)+35 = 31.16 seconds.

f) We now have X = 24. We need to first calculate the z-score. The z-score for X = 24 is z=(24-35)/3 = -9/3 = -3. If we look up -3 in the unit normal table, probability in the tail for z=-3 is 0.0027. In other words, only 0.27% of individuals would run faster than 24 sec. Based on this low probability, it would be unusual.