Business Statistics and Research Methods Questions and Answers

Answer: C

The probability is 24/55.

The box holds 11 balls, so two can be drawn in

\[ \binom{11}{2} = \frac{11 \times 10}{2} = 55 \]

ways. One white and one red can be chosen in 8 × 3 = 24 ways, so

\[ P = \frac{8 \times 3}{55} = \frac{24}{55} \]

28/55 is the probability of two white balls (28 ways) and 3/55 that of two red balls (3 ways), so both are answers to different questions.

Answer: B

The hypothesis tested is the null hypothesis.

In significance testing, the null hypothesis (of no difference or no effect) is the statement actually tested. The sample evidence leads either to rejecting it or to failing to reject it, and the alternative hypothesis is accepted only by rejecting the null.

So the alternative is not tested directly, both are not tested together, and a descriptive hypothesis is a different kind of statement altogether.

Answer: A

The coefficient of correlation varies between ± 1.

Karl Pearson's coefficient r always lies between -1 (perfect negative correlation) and +1 (perfect positive correlation), with 0 meaning no linear correlation.

So it is not without limit and can never be more than 1. 'Can be less than 1' is true of most values but does not state the property, since r also cannot be less than -1.

Answer: C

The factor that does not usually affect the width is population size.

The width of a confidence interval for a mean depends on the confidence level desired (through the critical value), the variability in the population (the standard deviation), and the sample size (through the standard error, which shrinks as the sample grows).

Population size matters only when the sample is a large fraction of a finite population, through the finite population correction, so it does not usually affect the width.

Answer: B

The necessary but not sufficient condition is reliability.

Goodness of measures is judged by reliability (consistency) and validity (measuring what is intended). A measure cannot be valid unless it is reliable, so reliability is necessary; but a measure can be consistently wrong, so reliability alone does not make it valid and is not sufficient.

Validity is the end that reliability serves, not the necessary condition. Measurement and sensitivity are not the conditions this standard statement refers to.

Answer: C

The false statement is 1 - α is called power of the test.

α is the probability of a Type-I error (rejecting a true null hypothesis), and β is the probability of a Type-II error (accepting a false null hypothesis). The power of a test is the probability of correctly rejecting a false null hypothesis, which is 1 - β.

1 - α is the confidence level, the probability of correctly accepting a true null hypothesis, not the power. The other three statements are correct.

Answer: A

The non-probability design is the quota sample.

In quota sampling, the researcher fixes quotas for categories such as age or sex and then picks respondents by judgement or convenience until each quota is filled. Selection is not random, so units do not have a known chance of selection.

Stratified, cluster and systematic random sampling all select units by a random process with known probabilities, so they are probability designs.

Answer: B

The quartile deviation is \(\frac{Q_{3} - Q_{1}}{2}\).

The quartile deviation, or semi-interquartile range, is half the distance between the upper quartile Q₃ and the lower quartile Q₁. It measures the spread of the middle 50 percent of the data and is not affected by extreme values.

\[ \text{Q.D.} = \frac{Q_{3} - Q_{1}}{2} \]

Q₁ - Q₂ uses the median instead of the upper quartile, Q₁ - Q₃ gives a negative value, and putting 2 in the numerator inverts the formula.

Answer: D

The answer is that the sampling distribution of the mean approaches normal as the sample size increases.

The central limit theorem states that, whatever the shape of the population (provided it has a finite variance), the distribution of the sample mean becomes closer and closer to a normal distribution as the sample size grows, with mean equal to the population mean and standard error σ/√n.

It is not 'always normal': for small samples from a non-normal population it is not. Even for large samples the theorem gives an approximation that improves with n, not exact normality, and no fixed size such as 1,000 is required; samples of about 30 are usually treated as large enough.

Answer: A

Cause and effect is studied through regression.

Regression analysis treats one variable as dependent and another as independent and estimates how the dependent variable changes with the independent one, giving a functional relationship that can be used for prediction. It therefore assumes a direction of influence, from cause to effect.

Correlation measures only the degree of association between two variables and makes no assumption about which causes which. Probability and the coefficient of variation do not deal with relationships between variables at all.