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A binomial probability works for situations with two possible outcomes, such as a 'success' or 'failure.' In our example, having a red car is considered a 'success,' while not having a red car is a 'failure.' The binomial distribution is governed by two parameters: the number of trials, denoted as \(n\), and the probability of success in each trial, denoted as \(p\).

In the given exercise, the number of respondents (or trials) is \(n = 175\) and the probability of owning a red car according to DuPont is \(p = 0.06\). The binomial probability formula is:
\ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \
where \( k \) is the number of successes (red cars). The task is to find the probability of 17 or more successes in 175 trials. This involves calculating the cumulative probability.

Statistical significance helps us determine if a result is likely due to chance or if it's meaningful. To assess if the survey results deviate from DuPont's claim, we set a significance level (often denoted as \( \alpha \)), typically 0.05 for a 5% significance level.

Here's how it works:

• If the probability of getting 17 or more red cars (\(P(X\geq 17)\)) is less than 0.05, the results are significantly different from DuPont's rate.
• If \(P(X\geq 17)\) is greater than or equal to 0.05, the survey results are not significantly different from DuPont's claim, meaning there isn't strong evidence to suggest the proportion of red cars differs from DuPont's estimate.

The calculated probability of 17 or more red cars will help us interpret the survey results in the context of statistical significance.

The cumulative distribution function (CDF) of a binomial distribution sums up the probabilities of obtaining a number of successes up to a certain value. It is useful when we want to find the probability of observing a range of outcomes rather than a specific number.

In this problem, we need to find the probability of having 17 or more red cars out of 175 respondents. This is represented as \(P(X\geq 17)\), the sum of probabilities for every possible outcome from 17 to 175. To find this, we use the complementary rule:
\ P(X \geq 17) = 1 - P(X < 17) \.
The CDF used in this context is for all values less than 17. Today, we often use statistical software or a binomial calculator to find these probabilities quickly and accurately.

In the exercise, calculating \( P(X\geq 17) = 0.0353 \) shows the likelihood of seeing 17 or more red cars out of 175, providing the basis for deciding if the result is statistically significant against DuPont's claim.