Answer to Question #215815 in Statistics and Probability for Dylan

Question

The mean yield of a chemical process is being research by an engineer. From

previous experience with this process the standard deviation of yield is known

to be 3. He would like to be 99% confident that the estimate point should be

accurate within yield point with the value of one.

(i) Examine the error and then analyze how large a sample is necessary for

this research?

(ii) Suppose that an engineer reduced the sample size to 20. If it was found

that the sample mean is 10 and a standard deviation of 1.6, find a 99%

confidence interval for the mean yeild.

Answer 1

(i) The critical value for $\alpha=0.01$ is $z_c=z_{1-\alpha/2}=2.5758.$

$$SE=z_c\times\dfrac{\sigma}{\sqrt{n}}=1$$

$$n\geq(\dfrac{z_c\times\sigma}{1})^2$$

$$n\geq(2.5758(3))^2$$

$$n\geq60$$

(ii) The critical value for $\alpha=0.01$ is $z_c=z_{1-\alpha/2}=2.5758.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-z_c\times\dfrac{\sigma}{\sqrt{n}}, \bar{x}+z_c\times\dfrac{\sigma}{\sqrt{n}})$$

$$=(20-2.5758\times\dfrac{1.6}{\sqrt{10}}, 20+2.5758\times\dfrac{1.6}{\sqrt{10}})$$

$$=(18.697, 21.303)$$

Therefore, based on the data provided, the 99% confidence interval for the population mean is $18.697<\mu< 21.303,$ which indicates that we are 99% confident that the true population mean $\mu$ is contained by the interval $(18.697, 21.303).$