Exercise 3

1. A farmer has a flock of 237 crossbred lambs. The mean weight of the lambs is 35 kg with a standard deviation of 2 kg. If lambs between the weights of 33 to 39 kg are suitable for export, how many lambs in this flock could the farmer expect to be able to export?

prop <- pnorm(q = 39, mean = 35, sd = 2) - pnorm(q = 33, mean = 35, sd = 2) # 82%
prop * 237 # 194

## [1] 194.0069

2. When we consider the standard normal distribution, what percentage of the data lies between 1, 2 and 3 standard deviations?

pnorm(1) - pnorm(-1) # 68%

## [1] 0.6826895

pnorm(2) - pnorm(-2) # 95%

## [1] 0.9544997

pnorm(3) - pnorm(-3) # 99.7%

## [1] 0.9973002

With this exercise, you have just deduced the so-called 68–95–99.7 rule, stating that 68% of the values in a normal distribution lie within \(\pm\) one standard deviation, 95% of the values lie within \(\pm\) two standard deviations and 99.7% of the values lie within \(\pm\) three standard deviations.

2. The marks of a statistics course are normally distributed with a mean of 65 and a standard deviation of 11. If we pick one student of this class at random, what is the probability that the student scored

1. higher than 76 marks?

pnorm(q = 76, mean = 65, sd = 11, lower.tail = F) # ~ 16%

## [1] 0.1586553

2. between 43 and 76 marks?

pnorm(q = 76, mean = 65, sd = 11) - pnorm(q = 43, mean = 65, sd = 11) # 82%

## [1] 0.8185946

3. If the top 16% of the students received an A grade, what was the minimum mark for an A? (Round your answer to the nearest integer.)

qnorm(p = 0.16, mean = 65, sd = 11, lower.tail = F) # 76 marks

## [1] 75.93904

qnorm(p = 0.84, mean = 65, sd = 11) # same but ramped up from the lower tail

## [1] 75.93904

4. If 2964 students sat the exam, how many of them could be expected to score less than or equal to 32 marks?

pnorm(q = 32, mean = 65, sd = 11) * 2964

## [1] 4.001098