The Acme Candy Company claims that 60% of the jawbreakers it produces weigh more than .4 ounces. Suppose that 800 jawbreakers are selected at random form the production lines. Would it be unusual for this sample of 800 to contain 575 jawbreakers that weigh more than .4 ounces? Explain
yes
To determine whether it would be unusual for the sample of 800 jawbreakers to contain 575 jawbreakers weighing more than 0.4 ounces, we need to calculate the probability of observing this outcome.
First, we need to identify the distribution that best fits this situation. In this case, the appropriate distribution is the binomial distribution. The binomial distribution is used when we have a fixed number of independent trials, each with the same probability of success.
In this scenario, each jawbreaker can be considered an independent trial, where success is defined as a jawbreaker weighing more than 0.4 ounces. The probability of success, p, is given as 0.6.
The formula to calculate the probability of obtaining a specific number of successes in a binomial distribution is:
P(X = k) = (n C k) * (p^k) * (1-p)^(n-k),
where:
- P(X = k) is the probability of obtaining k successes,
- n is the total number of trials (800 in this case),
- k is the number of successful trials (575 in this case),
- (n C k) represents the number of combinations or ways to select k successes from n trials,
- p is the probability of success (0.6 in this case),
- (1-p) is the probability of failure (0.4 in this case).
Now, we can calculate the probability of getting exactly 575 jawbreakers weighing more than 0.4 ounces using the binomial distribution formula:
P(X = 575) = (800 C 575) * (0.6^575) * (0.4^(800-575))
To determine whether this outcome is unusual, we need to compare this probability with a predetermined threshold (often called the significance level). If the probability is below the threshold, we consider the outcome unusual.
It is common to choose a significance level of 0.05, which corresponds to a 5% chance of observing an outcome as extreme or more extreme than the one we are testing.
To calculate the probability, you can use a statistical software, an online binomial calculator, or a formula-based calculator specifically designed for probability calculations.
After calculating the probability, if it is below 0.05, we would conclude that it is unusual to observe 575 jawbreakers weighing more than 0.4 ounces in a sample of 800.
Remember, this analysis assumes that the production process is random, and the independence assumption of each jawbreaker holds true.