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Example exercises

Quantitative Methods 24/25

This document contains more question sets than the final exam. Questions in this document can be used as reference for the form and the difficulty level. It aims to have a full coverage to all topics taught in this module.

Question set 1: Probability table

A market research group specialize in providing assessments of the prospects of shopping

centre. In a city famous for tourism, the group categorize that 60% of the tourism shops are successful, and 40% are unsuccessful. The group assess products as good, fair and poor. In those successful shops, the products are assessed as good for 70%, fair for 20% and poor for 10%. For those shops turned out to be unsuccessful, the products are good for 20%, fair for 30%, poor for 50%.

Let event G = the quality of product as good

Let event S = the prospect of a store is successful

Answer the following questions:

1.1. Draw a probability table which include events G and event S with corresponding probabilities

1.2. For a randomly chosen store, what is the probability that products will be assessed as good?

1.3. If products for a store are assessed as good, what is the probability that it will be successful?

1.4. Are the events G and event S statistically independent?

1.5. Suppose that 5 store are chosen as random. What is the probability that at least one of them will be successful?

Question set 2: CI and Chebychev’s theorem

Suppose we have a random sample of size n = 100 from a population. We find that the sample mean =80, sample standard deviation s=30. The probability that the population mean μ falls into an interval between two points a and b is 90%. Point a and b is symmetric to the central point of distribution.

2.1 What is the terminology for the range between point a and b?

2.2 What is the terminology for the value of point a and point b?

2.3 If the population is normally distributed, select an appropriate value from table below and calculate values of a and b. Justify your choice from the table. Explain your calculation in all necessary intermediate steps.

2.4 (4 points) If the shape of the population distribution is unknown, How to find out the values of a and b? Explain your answer with expressions to define a and b.

Question set 3: Fuel economy problem

You are in a project studying the impact of technology innovation on fuel economy.

Specifically, you are studying the performance of vehicle, which is measured by variable milpgal as the miles per gallon the vehicle can travel. A data set is obtained from collecting information through 4 companies. Variable company contains numerical values 1, 2, 3, 4 for data collected from company 1, company 2, company 3 and company 4. Company 1 and 4 are leading companies in the industry and they have adopted an innovative technology before the period of data collection. Please answer the following questions:

3.1 Explain the meaning of the following command and describe what change can be expected to the data set after executing the command:

gen inno=(company==1 | company==4)

3.2 Next, a Box-Whisker graph has been created to study the distribution of subsets data. In the box-whisker plot below, the dash line denoted with mean_ 1 refers to the mean of first subset. the solid line denoted with mean_2 refers to the mean of the second subset. A histogram has also been created to show the distribution of the same subsets.

Do you expected the two distribution of subset data to be normally distributed? Justify your answer by using evidence in the box-whisker plot and histogram.

You start the investigation on the impact of variable price by using the following Stata commands. The output is also the following:

4.1 Explain what the Stata command shown above is trying to achieve gen Hprice=(price>r(mean))

and predict the range of values contained in variable Hprice.

Then you perform a hypothesis testing on the subsection data created from question 1. It is assumed that the variances of populations for both subsections equal.

4.2 What is the H0  and H1  in the test implemented by command ttest milpgale, by(Hprice) as shown above? Formulate H0  and H1  and interpret them.

4.3 3 cases of hypothesis testing results are produced, as shown in the last 2 lines. Set significance level to 5%. What conclusion do you arrive at with referring to each of three cases of the test? Combining results from 3 sets of tests altogether, what is your conclusion?

4.4 Based on your conclusion in question 3, will you advise to include variable Hprice into the model to help explain the change in variable milpgal? Justify your answer.