Answer true or false to each of the statements in parts (a) and (b), and explain your reasoning. a. Two data sets that have identical frequency distri

Step 1:
a. The statement in part (a) is false. Two data sets that have identical frequency distributions may not necessarily have identical relative-frequency distributions.
To illustrate this, let's consider two data sets:
Data Set 1: 1, 2, 3, 4, 5
Data Set 2: 10, 20, 30, 40, 50
Both data sets have the same frequency distribution, as each value appears once. However, their relative-frequency distributions differ. The relative-frequency distribution for Data Set 1 is $\{0.2,0.2,0.2,0.2,0.2\}$, while the relative-frequency distribution for Data Set 2 is $\{0.1,0.1,0.1,0.1,0.1\}$. Thus, the statement is false.
Step 2:
b. The statement in part (b) is true. Two data sets that have identical relative-frequency distributions will have identical frequency distributions.
To demonstrate this, let's consider two data sets:
Data Set 1: 1, 2, 3, 4, 5
Data Set 2: 10, 20, 30, 40, 50
If we calculate the relative frequencies for both data sets, we obtain the same values: $\{0.2,0.2,0.2,0.2,0.2\}$ for both sets. From these relative frequencies, we can infer the frequency distributions for both sets. In this case, both data sets have the same frequency distribution: $\{1,1,1,1,1\}$. Therefore, the statement is true.
Step 3:
c. The relative-frequency distributions are better than frequency distributions for comparing two data sets because they allow for a more meaningful comparison that is not influenced by differences in sample sizes.
Frequency distributions provide information about the counts or frequencies of individual values in a data set. While they can be useful, they do not account for differences in sample sizes between data sets. Comparing frequency distributions directly may lead to misleading conclusions, as larger data sets will naturally have higher frequencies.
On the other hand, relative-frequency distributions express the frequencies as proportions or percentages relative to the total sample size. By doing so, they provide a standardized way of comparing data sets, regardless of their sample sizes. This allows for a fairer comparison that focuses on the relative occurrence of values within each data set.
In conclusion, relative-frequency distributions are preferred over frequency distributions when comparing two data sets because they provide a more accurate and unbiased representation of the data, eliminating the influence of sample size disparities.

a. $\text{False.}$ Two data sets that have identical frequency distributions may have different sample sizes, resulting in different relative-frequency distributions. Relative frequencies are calculated by dividing the frequency of each observation by the total sample size. Since the sample sizes can differ between the two data sets, their relative-frequency distributions will not be identical.
b. $\text{True.}$ Two data sets that have identical relative-frequency distributions necessarily have the same proportions for each observation. This implies that the frequencies of the observations must also be the same in both data sets, leading to identical frequency distributions.
c. Relative-frequency distributions are better than frequency distributions for comparing two data sets because they eliminate the influence of sample size. By focusing on proportions rather than raw frequencies, relative-frequency distributions allow for a direct comparison of the distributions, even when the sample sizes differ. This makes it easier to identify patterns and similarities between data sets without being affected by differences in scale.