91Ó°ÊÓ

The data that follow are from two random samples of 37 college males and 42 college females with respect to their commute times to college. $$\begin{array}{rrrrrrrrrrrrrrr}\text { Time } & -M & & & & & & & & & & & & \\\\\hline 15 & 12 & 30 & 15 & 10 & 23 & 20 & 13 & 25 & 20 & 15 & 20 & 23 & 15 & 20 \\\15 & 18 & 15 & 20 & 20 & 8 & 10 & 15 & 18 & 20 & 15 & 25 & 20 & 10 & 25 \\\18 & 18 & 20 & 27 & 25 & 20 & 7 & & & & & \\\\\hline\end{array}$$ $$\begin{array}{lllllllllllllll}\hline \text { Time } & -\mathrm{F} & & & & & & & & & & & & & \\\\\hline 32 & 15 & 20 & 35 & 45 & 20 & 10 & 5 & 35 & 25 & 14 & 25 & 28 & 35 & 30 \\\24 & 28 & 15 & 30 & 30 & 30 & 40 & 25 & 20 & 18 & 20 & 15 & 30 & 24 & 30 \\ 25 & 20 & 10 & 60 & 20 & 25 & 27 & 25 & 40 & 22 & 25 & 25 & & & \\\\\hline\end{array}$$ a. What are the populations of interest? b. Describe statistically the distribution of both the male and the female "commute time" data using at least the mean, standard deviation, and a histogram. c. Do the two sets of data represent dependent or independent samples? Explain why. d. If the two sample sizes are unequal, does this dictate dependent or independent samples? Explain your answer. e. If the two sample sizes are equal, does this dictate dependent or independent samples? Explain your answer.

Step by step solution

01

Identify Populations of Interest

The populations of interest in the given data are the college males and females. Here, the commute times of 37 college males and 42 college females are analyzed.

02

Describe the Statistical Distribution

The statistical distribution for the data needs to be calculated separately for both the male and female populations. This involves calculating the mean, standard deviation, and creating a histogram for each group. The mean is calculated by summing all the commute times and dividing it by the total number of records. The standard deviation is the square root of the average of the squared differences from the Mean. The histogram can be created by organizing the data into various intervals and counting the number of records that fall into each interval.

03

Determine the Dependency of the Samples

The two sets of data represent independent samples because they are collected from two different groups, namely male and female college students. The commuting time of a male student does not affect the commuting time of a female student, hence they are independent.

04

Impact of Unequal Sample Sizes

Unequal sample sizes do not dictate whether the samples are dependent or independent. Dependency is determined by the nature of the relationship between the data points, not by the size of the samples.

05

Impact of Equal Sample Sizes

Even if the sample sizes are equal, it does not dictate that the samples are dependent or independent. The determination of dependency or independency is based on the nature of data or whether the outcome of one sample affects the outcome of the other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population of Interest

In any statistical analysis, identifying the population of interest is crucial. This refers to the entire group of individuals or items that one wants to understand or measure. In this exercise, the populations of interest are the groups of college males and college females. These groups are selected to study their commute times. Each group is investigated separately to understand general patterns or specific behaviors within these populations. By identifying the populations of interest, we set the stage for further analysis and ensure that our conclusions are valid for these specific groups.

Statistical Distribution

Understanding the statistical distribution of data is essential in statistics. It involves describing how the data are spread or clustered. For the commute times of college males and females, we calculate measures such as the mean and standard deviation.

• **Mean**: This is the average commute time for each group, providing insight into typical commute durations. It's calculated by adding up all the times and dividing by the number of individuals.
• **Standard Deviation**: This measure indicates how much the commute times vary around the mean. A smaller standard deviation means times are closer to the average, while a larger one suggests more variability.
• **Histogram**: Visual representations like histograms show how commute times are distributed across different intervals, making it easier to spot patterns, such as the skewness or central tendencies of the data.

Independent Samples

Samples are independent if the selection or measurement of one sample does not influence the other. In the provided data for college males and females, the samples are independent because the commute time of one gender does not affect the other.
Each sample collects data separately and without any relationship to the other. Understanding the independence of samples is important when choosing the correct statistical tests and when analyzing the results, as certain methodologies assume sample independence.

Unequal Sample Sizes

In statistics, it is common to encounter unequal sample sizes, where the number of observations in one group differs from another. In the given problem, there are 37 male participants and 42 female participants.
It's crucial to understand that unequal sample sizes do not inherently dictate whether samples are dependent or independent. The dependency relates solely to whether the data points in one sample are affected or influenced by those in the other sample. Unequal sizes can introduce complexities in analysis but do not determine the relationship between samples.

Dependent Samples

Dependent samples occur when the data points or measurements in one sample are directly related to those in another. This relationship could be due to pairing, such as measurements before and after a treatment, or because each participant contributes a measure to both samples.
In this exercise with commute times of college males and females, the samples are independent, not dependent, since one group's data doesn't influence the other's.
Knowing whether samples are dependent or independent is critical when deciding which statistical methods to apply, ensuring that conclusions are drawn appropriately.