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 from the production lines. Would it
be unusual for this sample of 800 to contain 494 jawbreakers that weigh more than .4 ounces?
Well, finding 494 jawbreakers that weigh more than 0.4 ounces out of 800 selected at random would be unusual indeed. It's like finding a unicorn at a clown convention – not something you see every day! So, yeah, it would definitely be an unusual occurrence.
To determine whether it would be unusual for this sample of 800 jawbreakers to contain 494 jawbreakers that weigh more than 0.4 ounces, we need to calculate the probability using the binomial distribution.
The binomial distribution is used when we have a fixed number of trials (in this case, 800) and each trial results in a success (jawbreaker weighing more than 0.4 ounces) or a failure (jawbreaker weighing less than or equal to 0.4 ounces). The formula for the probability of getting exactly x successes in n trials is:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- P(x) is the probability of getting exactly x successes
- n is the total number of trials (800 in this case)
- nCx represents the number of combinations (also known as choose) of taking x objects from a set of n objects (written as n! / (x! * (n-x)!)
- p is the probability of success on a single trial (60% or 0.6)
- q is the probability of failure on a single trial (40% or 0.4)
Using these values, we can calculate the probability of getting 494 successes (jawbreakers weighing more than 0.4 ounces) out of 800 trials using the binomial distribution formula.
P(494) = (800C494) * (0.6^494) * (0.4^(800-494))
To determine whether this probability is unusual, we can compare it to a predefined threshold, often called the significance level or alpha level. A commonly used threshold is 0.05 or 5%.
If the calculated probability is less than the significance level (0.05), then it would be considered unusual. If the probability is greater than or equal to the significance level, then it would not be considered unusual.
To calculate the probability, we can use a statistical software package or online calculator that provides binomial distribution calculations.