A survey of 250 lobster fisherman found that they catch an average of 32 pounds of lobster per day with a standard deviation of four pounds. If a random sample of 30 lobster fisherman is selected, what is the probability that their average catch is less than 31.5 pounds?

n = 30

x = 31.5
μ = 32
SEm = SD/√n
SEm = 4/√30
SEm = 0.73
z = ( x - μ ) / SEm
z = (31.5-32)/0.73
z = -0.68
P(z < -0.68) = 0.2483

Thanks Kuai that really helped me out.

Z = (score-mean)/SEm

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

I am not understanding how to set this up. I have n=30 SD=4 xbar=250 after that I am lost

Well, let me tell you, these lobster fishermen sure sound like a lively bunch! Now, to answer your question about probabilities, we'll need some math-y stuff.

We're dealing with a sampling distribution, specifically the distribution of sample means. The mean of this distribution is equal to the population mean, which is 32 pounds, and the standard deviation is equal to the population standard deviation divided by the square root of the sample size, which in this case is 4 pounds divided by the square root of 30.

Now, let's find the probability that their average catch is less than 31.5 pounds. To do this, we'll need to calculate the z-score, which is defined as the difference between the sample mean and the population mean divided by the standard deviation.

So, the z-score is equal to (31.5 - 32) / (4 / sqrt(30)). And I'll let you do the math to find that value.

Once you have the z-score, you can consult a standard normal distribution table (or a calculator) to find the probability associated with that z-score.

Remember, probabilities are just like lobsters – sometimes they can be a bit tricky to catch, but with a little bit of patience and some math skills, you'll be able to wrangle that probability in no time!

To solve this problem, we need to consider the sample mean.

The sample mean, denoted as x̄ (pronounced "x-bar"), represents the average catch of the selected lobster fishermen. In this case, we want to find the probability that x̄ is less than 31.5 pounds.

To calculate this probability, we can use the concept of the sampling distribution of the sample mean. According to the Central Limit Theorem, for large enough sample sizes, the sampling distribution of the sample mean follows a normal distribution, regardless of the shape of the population.

Now, let's calculate the parameters of the sampling distribution.

The mean of the sampling distribution (also known as the population mean) can be found using the formula:
μ = μ, where μ represents the population mean (32 pounds in this case).

The standard deviation of the sampling distribution (also known as the standard error) can be calculated using the formula:
σ / √n, where σ represents the population standard deviation (4 pounds in this case), and n represents the sample size (30 in this case).

Using these formulas, we can find the mean (μ) and standard deviation (σ) of the sampling distribution.

μ = 32 pounds

σ = 4 pounds / √30 ≈ 0.7303 pounds (rounded to four decimal places)

Next, we need to calculate the z-score for the given value of x̄ (31.5 pounds). The z-score measures the number of standard deviations an individual value is from the mean.

The z-score can be calculated using the formula:
z = (x - μ) / σ, where x represents the given value, μ represents the mean, and σ represents the standard deviation.

z = (31.5 pounds - 32 pounds) / 0.7303 pounds ≈ -0.6861 (rounded to four decimal places)

Now, we can find the probability corresponding to this z-score using a standard normal distribution table or a statistical calculator. The probability represents the area under the normal curve to the left of the z-score.

Looking up the z-score of -0.6861 in a standard normal distribution table, we find that the probability is approximately 0.2457 (rounded to four decimal places).

Therefore, the probability that the average catch of the selected lobster fishermen is less than 31.5 pounds is approximately 0.2457, or 24.57%.