The average healthy female chicken will lay an average of 275 eggs per year with a population standard deviation of 12.5 eggs. The FRCC Biology club is conducting an experiment where each member will "adopt" one egg laying chicken for the year and observe their development and egg production. There are 45 members of the club. Use Excel functions to answer the following questions about the average of the [tex]$45$[/tex] chickens adopted by the club.
What is the probability that the average number of eggs laid by the club chickens is fewer than 260 eggs this year?
Answer (2)
The probability that the average number of eggs laid by the 45 club chickens is fewer than 260 eggs is approximately 3.89 x 10^-16, indicating this scenario is extremely unlikely. To arrive at this conclusion, we calculated the standard error of the mean, compute the Z-score, and found the corresponding probability. Therefore, finding such a low average egg production among these chickens is improbable.
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Calculate the standard error of the mean: σ x ˉ = 45 12.5 ≈ 1.863 .
Calculate the z-score: z = 1.863 260 − 275 ≈ − 8.050 .
Find the probability using the z-score: P ( Z < − 8.050 ) ≈ 3.89 × 1 0 − 16 .
The probability that the average number of eggs laid by the club chickens is fewer than 260 eggs this year is 3.89 × 1 0 − 16 .
Explanation
Understand the problem and provided data We are given the average egg production of a healthy female chicken (275 eggs per year) and the population standard deviation (12.5 eggs). A biology club with 45 members is observing the egg production of chickens they've adopted, and we want to find the probability that the average egg production of these 45 chickens is less than 260 eggs.
Calculate the standard error of the mean First, we need to calculate the standard error of the mean, which is a measure of how much the sample mean is likely to vary from the population mean. The formula for the standard error is: σ x ˉ = n σ where σ is the population standard deviation and n is the sample size. In this case, σ = 12.5 and n = 45 . So, σ x ˉ = 45 12.5 ≈ 1.863
Calculate the z-score Next, we calculate the z-score, which tells us how many standard errors away from the population mean our sample mean is. The formula for the z-score is: z = σ x ˉ x ˉ − μ where x ˉ is the sample mean and μ is the population mean. In this case, x ˉ = 260 and μ = 275 . So, z = 1.863 260 − 275 ≈ − 8.050
Find the probability using the z-score Now, we need to find the probability that the average number of eggs is less than 260. This is equivalent to finding the area under the standard normal curve to the left of our calculated z-score. Using a standard normal distribution table or a calculator, we find that the probability corresponding to a z-score of -8.050 is extremely small. The probability is approximately 3.89 × 1 0 − 16 .
State the final answer Therefore, the probability that the average number of eggs laid by the club chickens is fewer than 260 eggs this year is approximately 3.89 × 1 0 − 16 . This is an extremely small probability, indicating that it is highly unlikely for the average egg production to be this low.
Examples
Understanding probabilities like this can be useful in many real-world scenarios. For example, if a farmer wants to predict the yield of their crops, they can use the average yield from previous years and the standard deviation to calculate the probability of having a yield below a certain threshold. This can help them make informed decisions about irrigation, fertilization, and other farming practices. Similarly, in quality control, manufacturers can use probabilities to assess the likelihood of producing defective products.
