Question

There is a difference in sports preferences between men and women, according to a recent survey. Among the 10 most popular sports, men include competition- type sports-pool and billiards, basketball, and softball-whereas women include aerobics, running, hiking, and calisthenics. However, the top recreational activity for men was still the relaxing sport of fishing, with \(41 \%\) of those surveyed indicating that they had fished during the year. Suppose 180 randomly selected men are asked whether they had fished in the past year. a. What is the probability that fewer than 50 had fished? b. What is the probability that between 50 and 75 had fished? c. If the 180 men selected for the interview were selected by the marketing department of a sporting goods company based on information obtained from their mailing lists, what would you conclude about the reliability of their survey results?

Short answer

To calculate the probability that fewer than 50 men in the sample fished in the past year, we need to sum the probabilities of having 0, 1, 2, ..., 49 men who fished in the past year. So we need to compute: \(P(x < 50) = \sum_{x=0}^{49} \binom{180}{x} (0.41)^x (1-0.41)^{180-x}\) Calculate each probability and sum them. b. What is the probability that between 50 and 75 had fished? To calculate the probability that between 50 and 75 men in the sample fished in the past year, we need to sum the probabilities of having 50, 51, 52, ..., 75 men who fished in the past year. So we need to compute: \(P(50 \leq x \leq 75) = \sum_{x=50}^{75} \binom{180}{x} (0.41)^x (1-0.41)^{180-x}\) Calculate each probability and sum them. c. If the 180 men selected for the interview were selected by the marketing department of a sporting goods company based on information obtained from their mailing lists, what would you conclude about the reliability of their survey results? If the 180 men selected were from mailing lists of a sporting goods company, the sample may be biased as those men are already more likely to be engaged in recreational activities like fishing. Therefore, the survey results may not be representative of the general population, which indicates that the reliability of the survey results might be compromised.

Step-by-step solution

Calculate the probability that fewer than 50 had fished

To calculate the probability that fewer than 50 men in the sample fished in the past year, we need to sum the probabilities of having 0, 1, 2, ..., 49 men who fished in the past year. So we need to compute: \(P(x < 50) = \sum_{x=0}^{49} \binom{180}{x} (0.41)^x (1-0.41)^{180-x}\) Calculate each probability and sum them. b. What is the probability that between 50 and 75 had fished?

Calculate the probability that between 50 and 75 had fished

To calculate the probability that between 50 and 75 men in the sample fished in the past year, we need to sum the probabilities of having 50, 51, 52, ..., 75 men who fished in the past year. So we need to compute: \(P(50 \leq x \leq 75) = \sum_{x=50}^{75} \binom{180}{x} (0.41)^x (1-0.41)^{180-x}\) Calculate each probability and sum them. c. If the 180 men selected for the interview were selected by the marketing department of a sporting goods company based on information obtained from their mailing lists, what would you conclude about the reliability of their survey results?

Comment on the reliability of the survey results

If the 180 men selected were from mailing lists of a sporting goods company, the sample may be biased as those men are already more likely to be engaged in recreational activities like fishing. Therefore, the survey results may not be representative of the general population, which indicates that the reliability of the survey results might be compromised.

Key concepts

Understanding Binomial Distribution

The binomial distribution is a fundamental concept in probability, frequently used to model the number of successes in a fixed number of independent trials. In our exercise, when considering whether each man has fished or not in the last year, each man questioned represents a trial, and fishing is the 'success'.
For a scenario where 180 men are questioned, if 41% report having fished, it can be modeled using a binomial distribution with parameters: number of trials (n=180) and probability of success (p=0.41).

• The formula for the probability of exactly k successes ("men who fished" in this context) is given by:
  \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]
• In simple terms, this calculates how often we expect exactly k out of 180 to have fished given the 41% probability each man states he fished.

This concept allows us to compute probabilities for specific questions, just like determining fewer than 50 men fished, indicating how powerful a tool the binomial distribution can be in analyzing survey data.

Assessing Survey Reliability

Survey reliability refers to the degree to which survey outcomes reflect the true opinions or behaviors of the population.
In this exercise, we aim to understand if the results from a sample of men accurately reflect fishing habits among men in general.

• The reliability is contingent on various factors, including sampling methods, sample size, and bias.
• If the sample is truly random and adequately sized, the results are expected to be more reliable and representative of the general population.

However, survey reliability can be easily compromised by selection bias, as seen in this scenario, reinforcing the need for careful survey design to ensure that results represent broader population trends.
Understanding these elements is crucial for interpreting survey data and making informed decisions based on findings.

Recognizing Sample Bias

Sample bias arises when the sample selected for a survey does not accurately represent the entire population. This can lead to skewed results that do not truly reflect the population's preferences or behaviors.
In the given problem, if the men surveyed were chosen via a sporting goods company's mailing list, it's likely those individuals are already more inclined towards sporting activities, potentially inflating participation rates such as fishing.

• The bias occurs because the sample may have commonalities that are not present in the general population, affecting outcomes.
• To avoid this bias, it is important to ensure that the sampling method used is random and covers different segments of the population equitably.

Understanding and mitigating sample bias helps in designing better surveys and extracting more accurate and reliable data from the research conducted. It's a reminder of the importance of demographic diversity in sampling methods.

Exploring Sport Preferences

Sport preferences can vary greatly across different demographics, including gender, age, and cultural backgrounds.
In the problem given, the exercise highlights differences in sports preferences between men and women: men tend to favor competitive sports, while women may lean towards physical fitness activities.

• These preferences can be influenced by societal norms and personal interests.
• Recognizing these distinctions is vital for industries such as marketing and product development, helping tailor strategies that meet varied consumer needs.

Additionally, preferences may evolve over time with cultural shifts, advancements in sports technology, and changing lifestyle trends.
Understanding sport preferences not only provides insight into consumer behavior but also aids in the broader understanding of cultural and social dynamics.