Exercise 1

A chemical company markets a polymer used in the manufacture of microprocessors microprocessors and stored in a tank whose characteristic to be controlled is the viscosity viscosity; this one must be included between 75 and 95 to be able to market the polymer.

Four extractions were performed in different areas of the tank and led to the following sample led to the values of the sample

x1 = 78, x2 = 85, x3 = 91, x4 = 76.

The company needs to estimate the viscosity and also to know the accuracy of this this estimate. The viscosity follows a normal distribution N(, 2) with independence of the measured values xi.

1) It is assumed that the variability of the manufacturing process is constant and known with the population standard deviation = 5. What is the 95% confidence interval 95% confidence interval for can be provided to customers? Is this interval contained in the specification ([75,95])?

2) Same questions if the variance is no longer assumed to be constant and known.

Exercise 2

In a telephone information center, a statistical study showed that

that the wait (in seconds) before the call is initiated follows a normal distribution

of mean 18 and standard deviation 7.2.

After a reorganization of the service, a study is carried out to check whether there has been an

an improvement of the waiting time. Out of 100 calls,

a mean of 15.7824 was measured. It is assumed that the variance of the

population is the same as before the reorganization.

1. Calculate the p-value of a test to decide whether the waiting time has decreased since the

decreased since the reorganization of the service.

2. Can you tell if the waiting time has decreased if you consider a

risk threshold of 2%?

3. Same question as in 2), if we consider a risk threshold of 0.05%.

Exercise 3

A pharmaceutical company wants to study the potential side effects of a drug on the

on the cholesterol level of patients. One hundred volunteers are chosen to test the drug.

chosen to test the drug.

a) Before the experiment, the average cholesterol level of these volunteers is

2.020.2g/l. Since the average risk level of the population is 2g/l, verify that this sample is representative at the risk

that this sample is representative at 5% risk.

b) After one month of treatment, only 97 volunteers return for testing.

Their average cholesterol level has increased to 2.09g/l with a standard deviation

of 0.25g/l. Is the difference significant at the 5% risk?

Note: We consider here that the samples are large enough (100 and

97) to be able to consider that the measured rates follow normal distributions (the standard deviations of the samples can

standard deviations of the samples can therefore be taken as standard deviations of

populations).