Question

Americans are really getting away while on vacation. In fact, among small business owners, more than half \((51 \%)\) say they check in with the office at least once a day while on vacation; only \(27 \%\) say they cut the cord completely. \({ }^{7}\) If 20 small business owners are randomly selected, and we assume that exactly half check in with the office at least once a day, then \(n=20\) and \(p=.5 .\) Find the following probabilities. a. Exactly 16 say that they check in with the office at least once a day while on vacation. b. Between 15 and 18 (inclusive) say they check in with the office at least once a day while on vacation. c. Five or fewer say that they check in with the office at least once a day while on vacation. Would this be an unlikely occurrence?

Short answer

Answer: The probability that five or fewer small business owners check in with the office at least once a day while on vacation is approximately 0.0207. Since the value is close to 0, it means the probability of this happening is low, so we can say it's an unlikely occurrence.

Step-by-step solution

Understand the binomial distribution formula

The binomial distribution formula is given as \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\), where \(n\) is the sample size, \(k\) is the number of successes, and \(p\) is the probability of success.

Use the binomial distribution formula to find the probability for part a

For part (a), we need to find the probability that exactly 16 small business owners check in with the office at least once a day while on vacation. Here, \(n=20\) and \(k=16\). Plugging these values into the binomial formula, we have: \(P(X=16) = \binom{20}{16} (0.5)^{16} (1-0.5)^{20-16} = \binom{20}{16} (0.5)^{16} (0.5)^{4}\) Now, calculate and find the probability:\(P(X=16) \approx 0.0577\)

Use the binomial distribution formula to find the probability for part b

For part (b), we need to find the probability that between 15 and 18 (inclusive) small business owners check in with the office at least once a day while on vacation. Here, we need to find the sum of the probabilities for \(k = 15, 16,17,\) and \(18\): \(P(15 \leq X \leq 18) = P(X=15) + P(X=16) + P(X=17) + P(X=18)\) Now, use the binomial formula to find probabilities for each value of \(k\) and sum them up: \(P(15 \leq X \leq 18) \approx 0.2026\)

Use the binomial distribution formula to find the probability for part c

For part (c), we need to find the probability that five or fewer small business owners check in with the office at least once a day while on vacation. Here, we need to find the sum of probabilities for \(k = 0,1,2,3,4,\) and \(5\): \(P(X \leq 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)\) Now, use the binomial formula to find probabilities for each value of \(k\) and sum them up: \(P(X \leq 5) \approx 0.0207\) Now, we need to determine if this occurrence is unlikely. Since the value is close to 0, it means the probability of this happening is low, so we can say it's an unlikely occurrence.

Key concepts

Probability Calculation

Probability plays a crucial role in understanding and predicting various phenomena. In this scenario, it helps us determine how often certain vacation work habits occur among small business owners. The concept of calculating probabilities involves determining how likely an actual event is to happen. When we talk about binomial distribution in probability calculation, we're dealing with a specific type of distribution based on binary outcomes, such as success or failure.

In binomial probability, we use a formula to calculate the likelihood of a given number of successes happening in a series of trials. Here's the formula:

• \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \)

Where:

• \( n \) is the number of trials or sample size, which in this example is 20.
• \( k \) is the number of times the event of interest occurs, which varies in the sub-questions.
• \( p \) is the probability of success on any given trial, which here is set to 0.5.

When small business owners are surveyed, calculations are made regarding the number who maintain work connections during vacations using this formula. Each sub-part of the problem calculates the probability for different values of \( k \), helping us assess specific vacation work habits.

Small Business Owners

Small business owners play a fundamental role in the economy. With their ventures, they bring innovation, create jobs, and often work beyond regular hours. Within this context, their work habits, even when on vacation, become a topic of interest. Understanding how small business owners balance work and leisure can offer insights into stress levels, productivity, and the broader dynamics of managing a business.

• More than half check in with the office daily during vacations, showcasing their strong attachment and responsibility to their businesses.
• Only 27% cut ties completely when on vacation, reflecting on the nature of their commitment.

When we use statistical tools like binomial distribution to study small business owners, we delve into behaviors and expectations surrounding work-life balance. These results could affect how resources or support are provided to them, acknowledging the pressures they face while also valuing their downtime.

Vacation Work Habits

Vacation work habits refer to how individuals, especially professionals like small business owners, manage their work responsibilities while taking time off. Despite the purpose of vacations being to relax and disconnect, modern communication tools often blur those lines.
This exercise illustrates the likelihood of different vacation work habits occurring among small business owners. Some key observations:

• Many choose to remain partially connected, with 51% checking in at least once a day.
• This trend highlights a potential inability to fully disconnect, possibly due to technology or work culture pressures.
• A smaller fraction, 27%, manage to abstain completely from checking in.

Using probability calculations like in the exercise, we identify how common non-disconnection is among vacationers. Analyzing cases where up to five out of the 20 business owners stick to their work routines suggests it's rare, hence labeled unlikely. Such conclusions assist in understanding owner behaviors and might prompt reassessments of how vacations are perceived and taken by small business owners.