Business Statistics and Research Methods Questions and Answers

Answer: C

The percentage of weeks with no accidents is 13.53%.

For a Poisson distribution with mean m:

\[ \begin{aligned} P(X = x) &= \frac{e^{-m} m^{x}}{x!} \\ P(X = 0) &= e^{-2} = 0.1353 \end{aligned} \]

So 13.53% of weeks are expected to be accident-free.

27.06% is \(2e^{-2}\), the probability of exactly one (or exactly two) accidents, not of none. 6.77% is half of the correct figure, and 7.39% is \(e^{2}\) misread as a percentage.

Answer: A

The level of significance is the probability of rejecting \(H_{0}\) when it is true.

This is the probability of a Type I error, written \(\alpha\). A 5% level of significance means the test is designed so that a true null hypothesis is wrongly rejected 5 times in 100.

Rejecting \(H_{0}\) when it is false is a correct decision whose probability is the power of the test. Accepting a true \(H_{0}\) is also correct, with probability \(1 - \alpha\). Accepting \(H_{0}\) when it is false is a Type II error, with probability \(\beta\).

Answer: A

The standard error is 1/2.

The standard error of the mean is the population standard deviation divided by the square root of the sample size:

\[ \begin{aligned} SE(\bar{x}) &= \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{100}} \\ &= \frac{5}{10} = \frac{1}{2} \end{aligned} \]

The mean of 50 plays no part. 1/20 is what you get by dividing 5 by n instead of by its square root, and 1/100 and \(1/\sqrt{2}\) do not follow from the formula at all.

Answer: D

Such research is descriptive research.

Descriptive research includes surveys and fact-finding enquiries of different kinds. Its purpose is to describe the state of affairs as it exists, for example consumer preferences or the frequency of shopping, without the researcher controlling the variables.

Analytical research uses facts or information already available and analyses them to evaluate the material. Applied research aims at solving an immediate practical problem, and conceptual research deals with abstract ideas or theory. None of these is defined by surveys and fact finding.

Answer: C

The coefficient of determination is 0.49.

The coefficient of determination is the square of the correlation coefficient:

\[ r^{2} = (0.70)^{2} = 0.49 \]

It means 49% of the variation in y is explained by its linear relationship with x.

0.70 is r itself, 0.35 halves r instead of squaring it, and -0.70 is impossible because a square cannot be negative; the coefficient of determination always lies between 0 and 1.

Answer: B

The main emphasis falls on (i), (ii), (iii): the methods employed, the assumptions made, and the detailed presentation of findings.

A technical report is written for specialists who need to judge the research, so it sets out the methods used, the assumptions made in the course of the study, and a detailed presentation of the findings with their limitations and supporting data.

Simplicity and attractiveness (iv, v) are the emphasis of a popular report, written for a general reader in plain language with an appealing layout. Any option including them describes the wrong kind of report.

Answer: B

The technique is network analysis.

Network analysis represents a project as a network of activities and events, showing their sequence and dependence. Its best-known forms, PERT and CPM, find the critical path, the longest chain of activities that fixes the project's duration, and are used to plan, schedule resources for and control complex projects such as construction or product launches.

Competition analysis studies rivals, financial analysis studies financial statements and returns, and internal analysis assesses a firm's own strengths and weaknesses. None of them is a project scheduling technique.

Answer: B

The geometric mean is zero.

The geometric mean of n values is the nth root of their product. If any observation is zero, the product is zero, and so is its root:

\[ G = \sqrt[n]{x_1 x_2 \cdots 0 \cdots x_n} = 0 \]

The result is neither negative nor positive. Because a zero makes the measure uninformative, the geometric mean is said to be unsuitable for such data, which is why 'cannot be calculated' is a tempting distractor, but the value itself is zero.

Answer: B

Karl Pearson's coefficient of skewness lies between -3 and +3.

Pearson's coefficient is

\[ Sk_p = \frac{3(\text{Mean} - \text{Median})}{\sigma} \]

and since the mean and median cannot differ by more than one standard deviation, it lies between -3 and +3, though in practice it rarely goes beyond -1 and +1.

The limits -1 to +1 belong to Bowley's coefficient of skewness, which is the trap; -2 to +2 and -4 to +4 are not the bounds of either measure.

Answer: D

Regression analysis studies the causal relationship between two or more variables (printed in the paper as 'casual').

Regression estimates how a dependent variable changes in response to one or more independent variables and expresses that relationship as an equation used for prediction. It treats one variable as depending on the others, which is the cause-and-effect reading the key takes.

The degree of association between variables is measured by correlation, which is the trap. Data variability is measured by dispersion, and finding the best alternative is an optimisation problem.