According To A Survey By Paint Manufacturer DuPont: 22% Of A

Question

According To A Survey By Paint Manufacturer Dupont 22 Of All Cars I According to a survey by paint manufacturer, DuPont, 22% of all cars in the United States are red. Suppose 20 cars are randomly selected and the number of red cars are recorded. Round probabilities to 4 decimal places. Explain why this is a binomial experiment. Find and interpret the probability that exactly 6 cars are red. Find and interpret the probability that fewer than 6 cars are red. Find and interpret the probability that at least 6 cars are red. Compute the mean and standard deviation of the binomial random variable. Please show work.

Dr. Jack HW Helper · Accepted Answer

The scenario described involves analyzing the number of red cars out of a sample of 20 cars, with a fixed probability of any individual car being red (22%). This setup exemplifies a binomial experiment because it satisfies the key conditions: a fixed number of independent trials (20 cars), each trial has only two possible outcomes (red or not red), the probability of success (car being red) is constant at 0.22, and the trials are independent. Given these characteristics, the binomial probability model is appropriate for calculating various probabilities regarding the number of red cars. The binomial distribution is characterized by two parameters: n (number of trials) and p (probability of success on each trial). In this scenario, n = 20 and p = 0.22. The probability of exactly k successes (red cars) is given by the binomial probability formula: $$ P(X = k) = \binom{n}{k} p^{k} (1 - p)^{n - k} $$ where $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose k successes from n trials. Probability that exactly 6 cars are red Calculating $P(X=6)$: $$ P(X=6) = \binom{20}{6} (0.22)^6 (0.78)^{14} $$ Using a calculator or statistical software: $$ \binom{20}{6} = 38,760 $$ $$ P(X=6) ≈ 38,760 \times (0.22)^6 \times (0.78)^{14} $$ $$ ≈ 38,760 \times 0.0001139 \times 0.1094 $$ $$ ≈ 38,760 \times 0.00001245 ≈ 0.4826 $$ Therefore, the probability that exactly 6 cars are red is approximately 0.4826. Interpreted, there's about a 48.26% chance that exactly 6 out of 20 randomly selected cars are red, given that the probability of any individual car being red is 22%. Probability that fewer than 6 cars are red This probability is the sum of the probabilities that 0, 1, 2, 3, 4, or 5 cars are red: $$ P(X $$ Calculating each term: $$ P(X=k) = \binom{20}{k} (0.22)^k (0.78)^{20 - k} $$ Using statistical software or binomial tables, the cumulative probability \(P(X $$ P(X $$ Interpreted, there is a roughly 69.11% chance that fewer than 6 cars out of the 20

According To A Survey By Paint Manufacturer DuPont: 22% Of A

According To A Survey By Paint Manufacturer Dupont 22 Of All Cars I

Paper For Above instruction

Probability that exactly 6 cars are red

Probability that fewer than 6 cars are red

Probability that at least 6 cars are red

Mean and Standard Deviation of the Binomial Random Variable

Conclusion

References