Answer to Question #111363 in Statistics and Probability for Anonymus

Question

a. In a butter-packing plant, the quality of butter packed in a day using a certain type of machine is normally distributed with mean and variance 4. On a particular day, 12 packets of butter were taken at random from this production line and their masses, measured in grams, were:
9.5 11.2 9.9 10.9 10.1 10.9
9.5 10.6 11.1 9.8 10.2 11.0
Find a 97% confidence interval for the mean mass produced by this machine.
Explain the answer.

b. What sample size would be required to estimate the population mean for a large file of invoices of a multinational company to within RM0.50 with 95% confidence, given that the estimated standard deviation of the value of the invoices is RM6?

Answer 1

$a)\hat{\mu}\approx 10.39\text{ --- point estimate of population mean}.\\ \sigma^2=4.\\ N=12.\\ \alpha=0.03.\\ z_{\alpha/2}=2.17.\\ \mu=\hat{\mu}\pm [z_{\alpha/2}\frac{\sigma}{\sqrt{N}}].\\ \mu=10.39\pm[2.17{\frac{2}{\sqrt{12}}}].\\ (9.14;11.64)\text{ --- our confidence interval}.\\ \text{We can say that this CI covers population mean with probability } 0.97.\\ b)\sigma=6.\\ \alpha=0.05.\\ z_{\alpha/2}=1.96.\\ z_{\alpha/2}\frac{\sigma}{\sqrt{N}}=0.5.\\ N=(\frac{z_{\alpha/2}\sigma}{0.5})^2.\\ N\approx 554.$