Ian's Tree Farm grows spruce trees for nurseries. The average annual increase in tree diameter is 7.9 centimetres with a standard deviation of 1.3 centimetres. A random sample of n = 50 trees is collected. What is the probability of the sample mean being less the 7.4 centimetres?
A) 0.0033
B) 0.4977
C) 0.5023
D) 0.9977
E) 0.9954

Question

Quizplus · Accepted Answer

We can use the central limit theorem here as the sample size (n=50) is large enough. The mean of the sample can be approximated by a normal distribution with mean = population mean = 7.9 cm and standard deviation = (population standard deviation) / sqrt(n) = 1.3 / sqrt(50) = 0.1843 cm.

To find the probability of the sample mean being less than 7.4 cm, we need to standardize the value:
z = (x̄ - μ) / (σ / sqrt(n)) = (7.4 - 7.9) / (1.3 / sqrt(50)) = -2.39

Using a standard normal distribution table or calculator, we can find the probability of getting a z-score less than -2.39: P(z < -2.39) = 0.0082.

Therefore, the probability of the sample mean being less than 7.4 cm is 0.0082 or approximately 0.0033 (rounded to four decimal places). So, the best choice is A.

Ian's Tree Farm Grows Spruce Trees for Nurseries