Interpretation of Data Questions and Answers

Answer: D

The most thickly populated state is D.

Population density is population divided by area. One lakh is 100,000:

\[ \begin{aligned} A &: 1{,}600{,}000 / 80{,}000 = 20 \\ B &: 1{,}000{,}000 / 50{,}000 = 20 \\ C &: 400{,}000 / 10{,}000 = 40 \\ D &: 300{,}000 / 5{,}000 = 60 \end{aligned} \]

D has 60 persons per square km, the highest. A has the largest population but also by far the largest area, so it is as thinly populated as B at 20 per square km, and C is in the middle at 40.

Answer: B

The appropriate measure is the median.

The mean uses the value of every score, so a few extreme scores pull it towards them. The median is the middle value of the ordered data and depends only on position, so it is unaffected by how extreme the end values are. For skewed data such as incomes it is the preferred average.

The mode is the most frequent value and can be unstable or multiple, and the range is a measure of dispersion, not of central tendency.

Answer: C

The mean will not change.

\[ \begin{aligned} \text{total} &= 12 \times 40 = 480 \\ \text{removed} &= 20 + 60 = 80 \\ \text{new mean} &= \frac{480 - 80}{10} = 40 \end{aligned} \]

The two values removed average exactly 40, the same as the overall mean, so taking them out leaves the mean unchanged. The data are sufficient, because only the total is needed, not each observation.

Answer: B

The appropriate measure is the quartile deviation.

The quartile deviation is half the distance between the first and third quartiles, so it depends only on the middle 50% of the scores and is not affected by extreme values at either end.

The range is decided entirely by the two extreme scores, and the average deviation and standard deviation use every score, so extreme values distort all three.

For the same reason the median, rather than the mean, is the preferred average for such skewed data.

Answer: A

The F-test is used in ANOVA.

Analysis of variance tests whether the means of three or more groups differ. It compares the variance between the group means with the variance within the groups, and their ratio is the F statistic, with \(s^{2}\) standing for each variance:

\[ F = \frac{s^{2}_{\text{between}}}{s^{2}_{\text{within}}} \]

A large F shows that the groups differ more than chance would allow. The t-test and Z-test compare two means at a time, and the chi-square test is used for frequency data, testing association or goodness of fit, not for comparing means through variance.

Answer: A

A Type-I error occurs when the null hypothesis is rejected even if it is true.

In hypothesis testing, a Type I error is a false positive: the researcher concludes there is an effect or difference when there is none. Its probability is the significance level, alpha, usually set at 0.05.

Accepting a false null hypothesis is the other mistake, a Type II error (a false negative), whose probability is beta. The options about rejecting the alternative hypothesis, or both hypotheses together, do not describe either standard error.

Answer: C

The lowest level scale is the nominal scale.

Stevens's four levels of measurement, from lowest to highest, are nominal, ordinal, interval and ratio. A nominal scale only names or classifies, such as gender, religion or blood group. The numbers attached have no order or size, so only counting and the mode can be used.

The ordinal scale adds rank order, the interval scale (printed 'Internal scale') adds equal distances between values, and the ratio scale adds a true zero, which makes it the highest level.

Answer: D

It indicates confidence about generalizing their finding from the sample to the population.

A test of significance asks how likely it is that a result found in the sample, such as a difference or correlation, arose by chance from sampling. A significant result, for example at the 5 per cent level, lets the researcher infer with stated confidence that the result holds in the population from which the sample was drawn.

Inter-coder reliability is measured with agreement statistics such as Cohen's kappa, and understanding bivariate and multivariate analysis is a matter of method, not of testing. Passing the driving test is an unrelated distractor.

Answer: D

Mean, median and mode are measures of central tendency.

Each gives a single typical value around which the data cluster. The mean is the arithmetic average, the median is the middle value when the data are arranged in order, and the mode is the most frequent value.

Measures of deviation (dispersion), such as range, standard deviation and variance, describe the spread of data. Measures of association, such as correlation coefficients, describe the relationship between two variables, and ways of sampling, such as random or stratified sampling, are methods of selecting cases, not summary statistics.

Answer: C

The curve is a Hyperbola.

For every point the product XY is the same: 1 x 60 = 2 x 30 = 3 x 20 = 4 x 15 = 5 x 12 = 60. So the points lie on XY = 60, that is \(Y = \frac{60}{X}\), a rectangular hyperbola: Y is inversely proportional to X.

A straight line needs equal changes in Y for equal steps in X (here they are 30, 10, 5, 3). Y falls but not by a constant ratio (30/60 = 0.5, 20/30 = 0.67), so it is not exponential, and a parabola would turn upward again.