Question

How long is your arm compared with your hand size? Put your right thumb at your left shoulder bone, stretch your hand open wide, and extend your hand down your arm. Put your thumb at the place where your little finger is, and extend down the arm again. Repeat this a third time. Now your little finger will probably have reached the back of your left hand. If the fourth hand width goes past the end of your middle finger, turn your hand sideways and count finger widths to get there. a) How many hand and finger widths is your arm? b) Suppose you repeat your measurement 10 times and average your results. What parameter would this average estimate? What is the population? c) Suppose you now collect arm lengths measured in this way from 9 friends and average these 10 measurements. What is the population now? What parameter would this average estimate? d) Do you think these 10 arm lengths are likely to be representative of the population of \(\mathrm{arm}\) lengths in your community? In the country? Why or why not?

Short answer

a) Count hand and finger widths. b) Estimates your personal arm length parameter. c) Population: 10 people's arm lengths; average estimates group arm length. d) Likely not representative.

Step-by-step solution

Determine Hand Measurements

Begin by measuring the number of times your open hand fits along your arm from shoulder to wrist. Each complete span from thumb to little finger across the arm is considered a hand width.

Measure Extra Finger Widths

If, after using hand widths, the span of your arm exceeds these measurements, turn your hand sideways and count the additional finger widths needed to complete the measurement.

Calculate Total Widths

Add the number of hand widths and any extra finger widths together to get the total width measurement of your arm.

Answer Part (a)

The total number of hand and finger widths is the length of your arm when using this method. Record this total number.

Define Parameters for Part (b)

If you repeat the measurement 10 times and calculate an average, this average estimates the parameter of your personal arm length based on hand width measurements; the population consists of multiple trials measuring a single individual's arm.

Define New Population for Part (c)

With measurements from 9 friends plus your own, the population becomes the set of arm lengths of the 10 people measured. The average now estimates the average arm length for this small group.

Evaluate Representativeness for Part (d)

Ten arm lengths may not be representative of the wider population. This sample size is too small and non-random, limiting its ability to reflect the distribution of arm lengths in the larger community or country.

Key concepts

Measurement Techniques

In the exercise, you're using a simple and practical measurement technique to gauge your arm's length. This involves placing your thumb at one end, stretching your hand fully, and measuring the span it reaches on your arm. Each span equals one hand width. This technique is casual and can vary between individuals based on hand size. For more precision, especially if comparing measurements with others, using a ruler or measuring tape could provide consistent results. However, this exercise aims to give you an understanding of approximate measurement methods, rather than high-precision tools.
When measuring something like arm length, accuracy can depend on:

• The consistency of hand positioning during each measurement.
• The careful counting of overlapping parts such as finger widths if the span doesn't end neatly.

These techniques highlight the basics of physical measurement without complex instruments.

Sampling Methods

Sampling methods are crucial in collecting data that accurately reflects a larger group. In the exercise, you consider taking your measurements, then expanding to include those of your friends. Here, two types of sampling methods are indirectly referenced:

• Convenience Sampling: This is used when you gather data from people who are easily accessible or willing to participate. If you're measuring just your friends, it's an example of this method, but it may not provide a representative sample of a larger population.
• Random Sampling: For more accuracy in reflecting a community or larger group's characteristics, a sample should be random. This involves selecting participants such that every member of the population has an equal chance of being chosen, minimizing bias.

Choosing the appropriate sampling method can greatly influence the validity of conclusions drawn. For small, casual samples like the exercise suggests, convenience sampling may be adequate. However, for broader research, random sampling is often essential.

Population and Sample

Understanding the difference between a population and a sample is key to effective data analysis.

• Population: This refers to the entire group you are interested in studying. In the exercise’s context, when measuring your arm multiple times, each measurement acts as part of your own measuring population. When you include your friends, the population becomes the collective length of arms among you and your friends.
• Sample: A sample is a subset of the population used to draw inferences about the whole. In part (b), the repetitive measurements form a sample of your own arm length data. In part (c), the arm lengths of you and your 9 friends make up a sample from a possibly larger population you wish to study, like your community’s or even a nation's arm lengths.

Evaluating your sampling strategy helps determine whether your sample can generalize findings to the population, which impacts the reliability of your estimates.

Data Collection

Data collection in this exercise involves gathering measurements by repeating the hand span measurement method a set number of times for accuracy. This process is crucial to gaining insight into patterns or averages, as seen in the repeated measurements taken from each participant. Effective data collection strategies encompass:

• Consistency: Ensure each measurement follows the same method to maintain accuracy across your dataset.
• Replication: Repeating measurements multiple times helps to account for variability and can lead to more reliable data.

In this exercise, data from you and 9 friends is averaged to provide an estimate of average arm length. This collective data, if gathered carefully, can offer a sense of patterns, though may not represent a broader population without a larger, more diverse set of samples. It's important to differentiate between collecting small, informal data versus extensive, precise data when analyzing findings.