Question

The typical American family spends lots of time driving to and from various activities, and lots of time in the drive-thru lines at fast-food restaurants. There is a rising amount of evidence suggesting that we are beginning to burn out! In fact, in a study conducted for the Center for a New American Dream, Time magazine reports that \(60 \%\) of Americans felt pressure to work too much, and \(80 \%\) wished for more family time. \({ }^{7}\) Assume that these percentages are correct for all Americans, and that a random sample of 25 Americans is selected. a. Use Table 1 in Appendix I to find the probability that more than 20 felt pressure to work too much. b. Use the normal approximation to the binomial distribution to aproximate the probability in part a. Compare your answer with the exact value from part a. c. Use Table 1 in Appendix I to find the probability that between 15 and 20 (inclusive) wished for more family time. d. Use the normal approximation to the binomial distribution to approximate the probability in part c. Compare your answer with the exact value from part c.

Short answer

Answer: The probabilities found using the exact binomial distribution in parts a and c are different from those found using the normal approximation in parts b and d. The normal approximation provides a close but not exact estimate of the probability compared to the binomial distribution. The difference observed between the exact probabilities and their normal approximations is due to the continuous nature of the normal distribution, while the binomial distribution is discrete.

Step-by-step solution

Calculate the probability using binomial distribution (part a)

We are given that the percentage of Americans who felt pressure to work too much is p = 0.6, and the sample size n = 25. We are asked to find the probability that more than 20 felt pressure to work too much. We will use the binomial distribution formula to find the probability: P(X > 20). Since we need to find the probability for X > 20, we should actually calculate the complement probability P(X ≤ 20) and then subtract it from 1: P(X > 20) = 1 - P(X ≤ 20) According to Table 1 in Appendix I, we can find the probabilities for X = 0, 1, 2, ... , 20 and add them up, then subtract the result from 1.

Calculate the probability using normal approximation to the binomial distribution (part b)

Now, let's use the normal approximation to approximate the probability in part a. To use the normal approximation, we will first find the mean and standard deviation of the binomial distribution: Mean, μ = np = 25 * 0.6 = 15 Standard deviation, σ = sqrt(np(1-p)) = sqrt(25 * 0.6 * 0.4) = 2.449 Now, let's use the standard normal distribution with Z = (X - μ) / σ and find the probability for P(X > 20): P(X > 20) ≈ P(Z > (20 - 15) / 2.449) = P(Z > 2.04) Using the Z-table, we find that P(Z > 2.04) ≈ 0.0207. Compare this value with the exact value obtained in part a.

Calculate the probability using binomial distribution (part c)

We are given that the percentage of Americans who wished for more family time is p = 0.8, and the sample size n = 25. We are asked to find the probability that between 15 and 20 Americans (inclusive) wished for more family time. We will use the binomial distribution formula to find the probability: P(15 ≤ X ≤ 20). According to Table 1 in Appendix I, we can find the probabilities for X = 15, 16, 17, ... , 20 and add them up.

Calculate the probability using normal approximation to the binomial distribution (part d)

Now, let's use the normal approximation to approximate the probability in part c. To use the normal approximation, we will first find the mean and standard deviation of the binomial distribution: Mean, μ = np = 25 * 0.8 = 20 Standard deviation, σ = sqrt(np(1-p)) = sqrt(25 * 0.8 * 0.2) = 2 Now, let's use the standard normal distribution with Z = (X - μ) / σ and find the probability for P(15 ≤ X ≤ 20): P(15 ≤ X ≤ 20) ≈ P((15 - 20) / 2 ≤ Z ≤ (20 - 20) / 2) = P(-2.5 ≤ Z ≤ 0) Using the Z-table, we find that P(-2.5 ≤ Z ≤ 0) ≈ 0.4938. Compare this value with the exact value obtained in part c.

Key concepts

Normal Approximation to Binomial Distribution

The normal approximation to the binomial distribution is a useful technique that allows us to use the normal distribution to estimate the probabilities of a binomial variable under certain conditions. This approximation becomes increasingly accurate when the number of trials () and the probability of success ((p)) in a binomial distribution satisfy certain criteria, typically when both np and n(1-p) are greater than 5.

For the normal approximation, the mean ((μ)) of the binomial distribution is calculated as np, while the standard deviation ((σ)) is found by the square root of np(1-p). We then convert the binomial probabilities into z-scores, which correspond to the number of standard deviations an observation is from the mean. This allows us to use standard normal distribution tables or calculators to approximate the probability.

Application of Normal Approximation

In the given exercise, the normal approximation was used to estimate the probability that more than 20 Americans felt the pressure to work too much. This is especially handy when dealing with large sample sizes where using the binomial formula directly becomes unwieldy. The normal approximation simplifies calculations and is a key tool in statistical analysis for various real-world scenarios.

Probability Calculations

Probability calculations are fundamental to interpreting binomial distributions and involving foundational concepts of determining the likelihood of various outcomes. In the exercise, the probability that a particular number of individuals experienced a certain condition was calculated using both the binomial formula and the normal approximation, which represents two different probability calculation methods.

In the first method, using the binomial distribution, we calculate the exact probability of our event. This requires summing up the probabilities of all events equal to or less than (or greater than) our target value and then taking the complement if necessary. The second method, normal approximation, provides an easier but approximate calculation method by relating our binomial variable to the standardized z-scores of the normal distribution.

Importance of Precise Calculations

It's important to note that exact and approximate methods can yield different results. The step-by-step solution showcases that different methods may be compared to assess the precision of an approximation, which is important when conducting statistical analyses in real-life scenarios where precise probability calculations are crucial.

Binomial Distribution Applications

The binomial distribution has extensive applications in various fields including health studies, quality control, survey analysis, and more. Its central premise is to model the number of successes in a fixed number of independent trials of a binary experiment. For example, flipping a coin (success/failure), inspecting items for defects (pass/fail), and in the case of our exercise, assessing whether Americans feel overworked or want more family time.

Binomial distribution is versatile and powerful, making it particularly useful in these fields for its ability to handle discrete events that have two outcomes. In our scenario, it helps in the understanding of societal trends based on the probabilities of individual responses.

Real-world Example

Using the scenario of the exercise, a researcher could apply these methods to gauge public sentiment on work-life balance. This application could be implemented in designing work policies, marketing strategies, or even in political campaigns. The ability to approximate probabilities efficiently through different methods exemplifies the practicality of the binomial distribution and its adaptations, like the normal approximation, in real-world decision-making.