A marine biologist has determined that the weight of a full weight of 2000pounds and a standard deviation of 500 pounds. Determine the probability that the average weight of the next five sharks captured will be less than 2100lbs.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion from the Z score. This is the probability for one shark.
The probability for all events is found by multiplying the probabilities of the individual events.
To determine the probability that the average weight of the next five sharks captured will be less than 2100 lbs, we need to use the Central Limit Theorem.
1. First, we need to find the distribution of the sample mean. The population mean is given as 2000 lbs, and the standard deviation is 500 lbs.
The standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
Standard Error = Standard Deviation / sqrt(n)
Here, n is the sample size, which is 5 in this case.
Standard Error = 500 / sqrt(5)
≈ 223.61 lbs
2. Next, we need to convert the average weight of 2100 lbs into a z-score. The z-score is calculated using the formula:
z = (x - μ) / SE
Here, x is the average weight we want to find the probability for, μ is the population mean, and SE is the standard error.
z = (2100 - 2000) / 223.61
= 0.447
3. Using a z-table or a statistical software, we can find the probability associated with the z-score. Looking up the z-score of 0.447 in the z-table, we find the probability to be approximately 0.6736.
Therefore, the probability that the average weight of the next five sharks captured will be less than 2100 lbs is approximately 0.6736, or 67.36%.
To determine the probability that the average weight of the next five sharks captured will be less than 2100 lbs, we need to use the concept of the sampling distribution of the mean.
First, let's calculate the standard deviation of the sampling distribution of the mean. The standard deviation of the sampling distribution, often referred to as the standard error, is the standard deviation of the population divided by the square root of the sample size. In this case, the population standard deviation (σ) is 500 pounds, and since we are taking a sample of 5, the sample size (n) is 5. The standard error (SE) can be calculated as follows:
SE = σ / √n
SE = 500 / √5
SE ≈ 500 / 2.236 ≈ 223.6 pounds
Next, we need to convert the average weight of 2100 lbs into a standardized score (z-score) using the formula:
z = (x - μ) / SE
Where x is the value we want to convert (2100 lbs), μ is the population mean (2000 lbs), and SE is the standard error (223.6 lbs). Let's calculate the z-score:
z = (2100 - 2000) / 223.6
z ≈ 100 / 223.6 ≈ 0.447
Now, we need to find the probability associated with this z-score using a standard normal distribution table or a calculator. The probability represents the area under the curve to the left of the z-score.
Using a standard normal distribution table or a calculator, we can find that the probability corresponding to a z-score of 0.447 is approximately 0.673. This means that the probability that the average weight of the next five sharks captured will be less than 2100 lbs is approximately 0.673 or 67.3%.
Therefore, based on the given information, there is a approximately 67.3% probability that the average weight of the next five sharks captured will be less than 2100lbs.