Question

Problems 6 - 8: Suppose that the amount of coffee in each cup dispensed by a vending machine follows a normal distribution. A coffee vending machine is configured so that mean of all of the cups of coffee it dispenses is u = 12.0 ounces with a standard deviation o = 0.8 ounces. The mean of the distribution can be set by adjusting the filling machinery, while the standard deviation reflects the precision of the filling machinery. During a quality control check, a random sample of 16 cups is collected. The amount of coffee is each cup is measured and the sample mean is determined to be 11.5 ounces. 6) Assuming u = 12.0 ounces and o = 0.8 ounces, what is the probability that 16 randomly selected cups will have a sample mean less than 11.5 ounces? State your answer as decimal rounded to 4 places. 7) On the accompanying graph, label , X and shade the area that corresponds to the probability that 16 randomly selected cups will have a sample mean less than 11.5 ounces (note: shading should be proportional to actual area...) 8) Using your answer from the problem above, If 0.05 or smaller is considered unusual, would it be unusual to randomly select 16 cups that would have a sample mean less than 11.5 ounces? Explain.

Answer 1

Sample of size 16 will have the same mean 12 and standard deviation

0.8 0.8 = 0.2 n 16 4

Required probability is $P(\bar{X}<11.5)=P\left ( \frac{\bar{X}-12}{0.2}<\frac{11.5-12}{0.2} \right )=P(Z<-2.5)$ where .$Z\sim N(0,1)$ is standard normal

Consulting a standard normal table, we have $P(Z<-2.5)\approx 0.006$ is the required probability

$\blacksquare$