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Chapter 4: Problem 8

Can you raise one eyebrow at a time? Use the students in your statistics class (or a group of friends) to estimate the percentage of people who can raise one eyebrow at a time. How can your result be thought of as an estimate for the probability that a person chosen at random can raise one eyebrow at a time? Comment: National statistics indicate that about \(30 \%\) of Americans can raise one eyebrow at a time (see source in Problem 7)

Short Answer

Expert verified
Collect data, calculate the sample percentage, convert it to a probability, and compare it with the national statistic of 30%.

Step by step solution

01

Collect Data

Gather data by observing each student or friend and note whether they can raise one eyebrow at a time. Record these observations.
02

Calculate the Sample Percentage

Count the number of participants who can raise one eyebrow and divide that by the total number of participants. Multiply the result by 100 to convert it to a percentage.
03

Express as Probability

Use the percentage as an estimate for probability by dividing the percentage by 100, giving the probability in decimal form.
04

Compare with National Statistics

Note how the calculated probability compares to the national statistic of 30% who can raise one eyebrow, recognizing any differences or similarities.

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Key Concepts

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

Data Collection
Before we dive into estimating probabilities or making comparisons, we must first gather relevant data. In this exercise, the goal is to determine the percentage of people who can raise one eyebrow at a time. This begins with observing your friends or classmates. Simply ask each person in your chosen group to attempt raising one eyebrow individually.

Once each attempt is complete, record the results. Create a tally of how many successfully raised one eyebrow versus those who couldn鈥檛. The key here is accuracy in recording each observation to ensure your findings are as reliable as possible.
  • Ensure each participant is in a relaxed setting to avoid any performance anxiety.
  • Maintain a neutral expression when recording results to avoid influencing your participants.
  • Try to include a diverse group of people to help in making your sample more representative.
Sample Percentage
After collecting your data, the next step is to compute the sample percentage. This percentage will tell you more about your group's eyebrow-raising abilities. Begin by taking the total number of people who successfully raised one eyebrow and dividing it by the total number of participants you observed.

For instance, if 7 out of 20 people managed to do it, you compute the sample percentage as follows:
  • Formula: \[ \text{Sample Percentage} = \left( \frac{\text{Number of Successes}}{\text{Total Number of Participants}} \right) \times 100 \]
  • Example: \[ \text{Sample Percentage} = \left( \frac{7}{20} \right) \times 100 = 35\% \]

Having the percentage framed in this context helps you visualize the proportion of people in your sample that can actually raise one eyebrow.
Comparison with National Statistics
With your sample percentage in hand, the next logical step is to compare it with known national statistics. Nationally, it's stated that about 30% of Americans have the ability to raise a single eyebrow.

If your sample percentage mirrors this 30% close enough, it could suggest that your group's behavior is representative of the larger population.
  • Is your sample above 30%? You might be dealing with a more expressive group.
  • If below 30%, environmental factors or sample size could be influencing results.
  • Discrepancies can lead to intriguing questions about your sample's uniqueness.

Use these observations to understand how your limited sample might align or diverge from the broader population.
Observed Probability
Finally, your sample percentage can also be interpreted as an observed probability. Probability allows us to express the chance of an event happening, in this case, the ability to raise one eyebrow.

To convert your sample percentage to a probability, simply divide by 100. Returning to our example with a sample percentage of 35%, you would compute as follows:
  • Formula: \[ \text{Probability} = \frac{\text{Sample Percentage}}{100} \]
  • Example: \[ \text{Probability} = \frac{35}{100} = 0.35 \]

This means there's a 35% chance, or a probability of 0.35, that a randomly selected person from your group can indeed raise one eyebrow.

Understanding this observed probability helps to bridge individual observations back to broader statistical concepts.

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