In Exercises 25–28, find the probabilities and answer the questions.

Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.

d. If we randomly select five adults, is 1 a significantly low number who say that they were too young to get tattoos?

검증된 단계별 안내

Step 1: Identify the type of probability distribution involved. Since we are dealing with a fixed number of trials (5 adults), each with two possible outcomes (either they say they were too young or they don't), and the probability of success (saying they were too young) is constant at 20% (or 0.2), this is a binomial probability problem.

Step 2: Write the formula for the binomial probability distribution. The probability of exactly k successes in n trials is given by:

P (k) = (\frac{n!}{k! (n - k)!}) \times p^{k} \times (^{1 - p) n - k}

, where n is the number of trials, k is the number of successes, and p is the probability of success.

Step 3: Plug in the values for this problem. Here, n = 5 (the number of adults), k = 1 (the number of adults who say they were too young), and p = 0.2 (the probability of saying they were too young). Substitute these values into the formula to calculate the probability of exactly 1 success.

Step 4: Determine whether 1 is a significantly low number. To assess this, calculate the cumulative probability of 0 or 1 success (P(X ≤ 1)). This involves summing the probabilities for k = 0 and k = 1. Compare this cumulative probability to a significance threshold, such as 0.05, to decide if 1 is significantly low.

Step 5: Interpret the result. If the cumulative probability P(X ≤ 1) is less than 0.05, then 1 is considered a significantly low number. Otherwise, it is not. Use this interpretation to answer the question about whether 1 is significantly low.

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주요 개념

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Binomial Probability

Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is defined as an adult stating they were too young to get a tattoo, with a success probability of 20%. The binomial formula is used to calculate the probability of observing a certain number of successes (e.g., 1 adult) in a sample of five.

Significance Level

The significance level is a threshold used to determine whether a result is statistically significant. In this scenario, it helps assess whether the observed number of adults (1) who feel they were too young is significantly low compared to what would be expected based on the binomial distribution. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. In this case, we would set up a null hypothesis (that 1 adult is not significantly low) and an alternative hypothesis (that 1 adult is significantly low). By calculating the probability of observing 1 or fewer adults who feel too young, we can determine whether to reject the null hypothesis in favor of the alternative.