Question

The weights of the fish in a certain lake are normally distributed with a mean of 11 lb and a standard deviation of 6. if 4 fish are randomly selected, what is the probability that the mean weight will be between 8.6 and 14.6 lb? your answer should be a decimal rounded to the fourth decimal place.

Answer 1

To find the probability that the mean weight of 4 randomly selected fish is between 8.6 and 14.6 lb, we can use the Central Limit Theorem. The probability is approximately 0.7881.

Step-by-step explanation:

To find the probability that the mean weight of 4 randomly selected fish is between 8.6 and 14.6 lb, we can use the Central Limit Theorem. The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.

In this case, we have a normal distribution with a mean of 11 lb and a standard deviation of 6 lb. Since we are sampling 4 fish, the sample mean will also have a normal distribution with a mean of 11 lb and a standard deviation of 6/sqrt(4) = 3 lb.

To find the probability that the sample mean is between 8.6 and 14.6 lb, we can standardize the values using the z-score formula: z = (x - μ) / (σ / sqrt(n)). Plugging in the values, we have z1 = (8.6 - 11) / (3) = -0.8 and z2 = (14.6 - 11) / (3) = 1.2. We can then use a standard normal distribution table or calculator to find the probability between these z-scores.

Using a standard normal distribution table or calculator, we find that the probability that the mean weight is between 8.6 and 14.6 lb is approximately 0.7881.

Answer 2

Since the sample standard deviation is now known, we use the z-score test. The formula is given as:

z= (X – μ) / (s / sqrt(n))

Where,

X = sample mean = 8.6 lb to 14.6 lb

μ = population mean = 11 lb

s = population standard deviation = 6

n = sample size = 4

1st: Calculating for z when x = 8.6 lb

z = (8.6 – 11) / (6 / sqrt4)

z = - 0.8

Using the standard distribution table for z:

Probability (x = 8.6 lb) = 0.2119

2nd: Calculating for z when x = 14.6 lb

z = (14.6 – 11) / (6 / sqrt4)

z = 1.2

Using the standard distribution table for z:

Probability (x = 14.6 lb) = 0.8849

Therefore the probability that the mean weight will be between 8.6 and 14.6 lb:

Probability (8.6 ≤ x ≤ 14.6 ) = 0.8849 - 0.2119

Probability (8.6 ≤ x ≤ 14.6 ) = 0.673 (ANSWER)