6 marks) Below is the graph of a function:

Determine the numerical value of k so that f(x) is a density function. Problem

2. (22 marks) The machining process of the company TechNet tends to go wrong with time. Yesterday, the quality control technician randomly selected a sample of 24 parts and carefully measured the diameters.

The mean and standard deviation of the sample was found to be x 8.015 cm and s=0.037 cm, respectively. In addition, measurements performed in the past on a large number of similar parts suggested that the diameters follow a normal distribution.

(a) (2 marks) Determine a point estimate of the mean diameter, , of the parts made by this machining process.

(b) (4 marks) Find a 98% confidence estimate of the mean diameter, , of the parts made by this machining process.

(c) (2 marks) Interpret the values found in Part (b).

2 The production deadline being very tight, it has been decided that the process would not be adjusted; however, as a precaution, a random sample of 15 pieces has been measured today, which provided a mean of 8.027 cm and a standard deviation of 0.019 cm. In addition, it seems reasonable to assume the variances of parts produced yesterday and today to be equal. In the following, the significance level of 5% is considered.

(d) (8 marks) Can we conclude that the average diameter of the parts produced today and the average diameter of the parts produced yesterday are different?

(e) (6 marks) Do we have enough statistical evidence to claim that the mean diameter of the parts produced today is less than the mean diameter of the parts produced yesterday?