Business Statistics and Research Methods Questions and Answers

Answer: B

Random sampling is preferred because the results will be statistically valid.

In a random sample every member of the population has a known, usually equal, chance of selection. This removes the investigator's bias in choosing units, makes the sample representative on average, and allows the laws of probability to be used to estimate sampling error and draw valid inferences about the population.

It is not preferred for being easy to draw, since it needs a complete list of the population. Being popular is not a statistical reason, and it is not free from errors: sampling error remains, though it can be measured.

Answer: D

The one that is not an absolute measure is the coefficient of variation.

Absolute measures of dispersion are expressed in the units of the data, rupees or kilograms for instance. The coefficient of variation is the standard deviation divided by the mean, multiplied by 100, so the units cancel and it is a pure percentage. That makes it a relative measure, used to compare the variability of series with different units or averages.

The range, mean deviation and standard deviation are all absolute measures, expressed in the units of the original data.

Answer: C

The empirical formula is Mode = 3 Median - 2 Mean.

Karl Pearson observed that in a moderately skewed distribution the median lies about one-third of the way from the mean to the mode, which gives:

\[ \begin{aligned} \text{Mean} - \text{Mode} &= 3(\text{Mean} - \text{Median}) \\ \text{Mode} &= 3\,\text{Median} - 2\,\text{Mean} \end{aligned} \]

It is used when the mode is ill-defined. The option '2 Mean - 3 Median' is the same expression with the wrong sign, and the versions with 'Mean - 2 Median' or a plus sign do not follow from the relation at all.

Answer: B

A parameter is the population mean.

In statistics a parameter is a numerical measure that describes a characteristic of the population, such as the population mean (μ), standard deviation (σ) or proportion. Its counterpart computed from a sample is called a statistic and is used to estimate the parameter.

The sample mean is therefore a statistic, not a parameter. The size of the sample is chosen by the investigator, and the size of the population is a count of units rather than a measure describing the distribution of the characteristic studied, so neither is the parameter the question means.

Answer: B

The events are mutually exclusive.

On a single toss the coin cannot show both a head and a tail, so A and B cannot occur together. That is the definition of mutually exclusive events: \(P(A \cap B) = 0\).

They are not independent. For independent events the product rule must hold:

\[ \begin{aligned} P(A)P(B) &= \tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4} \\ P(A \cap B) &= 0 \ne \tfrac{1}{4} \end{aligned} \]

Indeed, knowing that A happened tells us for certain that B did not. Two events with non-zero probabilities can never be both mutually exclusive and independent, which rules out the third option, and the last option is wrong because they are mutually exclusive.

Answer: A

The equality of two population means is tested with the t test.

The two-sample t test compares the means of two independent samples. Its statistic divides the difference between the sample means by the standard error of that difference, and the result is compared with the t distribution with the appropriate degrees of freedom. For large samples the same test is usually done as a z test.

The F test compares two variances (and is used in analysis of variance for three or more means). The chi square test is used for goodness of fit, for independence in contingency tables and for a single variance. The Kolmogorov-Smirnov test compares whole distributions rather than means.

Answer: C

The sampling distribution is approximately normal when the sample size is sufficiently large.

By the central limit theorem, the sampling distribution of the mean (and of many other statistics) approaches a normal distribution as the sample size grows, whatever the shape of the population, provided its variance is finite. How large is large enough depends on the population; for most practical purposes a sample of 30 or more is taken as sufficient.

A small sample gives a normal sampling distribution only if the population itself is normal. 'Large' on its own is vaguer than the theorem's condition, which is that n is large enough for the approximation to hold, and 'optimum' sample size is a matter of cost and precision, not of the shape of the distribution.

Answer: D

Five percent corresponds to an angle of 18.0 degrees.

A pie diagram represents the whole, 100 per cent, by the full circle of 360 degrees, so each per cent is 3.6 degrees:

\[ \begin{aligned} \theta &= \frac{5}{100} \times 360^{\circ} \\ &= 18^{\circ} \end{aligned} \]

The other values are fractions of the correct angle: 9.0 is half of it, as if the circle were 180 degrees, 4.5 a quarter and 3.0 a sixth. None of them corresponds to 5 per cent of a full circle of 360 degrees.

Answer: C

The standard deviation of a binomial distribution is \(\sqrt{npq}\).

For a binomial distribution with n trials, probability of success p and probability of failure q = 1 - p, the mean and variance are:

\[ \begin{aligned} \mu &= np \\ \sigma^{2} &= npq \\ \sigma &= \sqrt{npq} \end{aligned} \]

So np is the mean, not the standard deviation, and npq is the variance; the standard deviation is its square root. The expression \(np^{2}q^{2}\) has no role in the binomial distribution. For example, with n = 100 and p = q = 0.5, the mean is 50, the variance 25 and the standard deviation 5.

Answer: A

The arithmetic mean will increase by 5.

The mean is affected by a change of origin. If every observation becomes x + 5, the sum rises by 5n, so

\[ \frac{\sum (x + 5)}{n} = \frac{\sum x}{n} + 5 = \bar{x} + 5 \]

The mean cannot stay unchanged or fall when every value rises, and the change is exactly 5 whatever the observations are. (Measures of dispersion such as the standard deviation, by contrast, are unaffected by a change of origin.)