The wildlife department has been feeding a special food to rainbow trout fingerlings in a pond. A sample of the weights of 40 trout revealed that the mean weight is 402.7 grams and the standatd deviation 8alE8 gr. What is the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams?
find p-value when
z >(x-mean)/(standard deviation)
by looking up in the z-score chart
To find the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams, we need to use the Central Limit Theorem.
The Central Limit Theorem states that for a large sample size, the distribution of the sample mean approximates a normal distribution, regardless of the shape of the population distribution.
In this case, since we have a sample size of 40, we can assume that the sample mean follows a normal distribution.
Now, in order to calculate the probability, we need to standardize the sample mean using the z-score formula:
z = (x - μ) / (σ / √n)
where:
- x is the sample mean (405.5 grams in this case)
- μ is the population mean (402.7 grams in this case)
- σ is the population standard deviation (8.8 grams in this case)
- n is the sample size (40 in this case)
Calculating the z-score:
z = (405.5 - 402.7) / (8.8 / √40)
z = 2.8 / (8.8 / √40)
z = 2.8 / (8.8 / 6.3246)
z = 2.8 / 1.3953
z ≈ 2.005
Now, we need to find the probability that the z-score is greater than 2.005. We can use a standard normal distribution table or calculator to find this probability.
Using a standard normal distribution table, we can find that the area to the right of a z-score of 2.005 is approximately 0.0228. This means that the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams is approximately 0.0228 or 2.28%.
So, there is a 2.28% probability that the mean weight for a sample of 40 trout exceeds 405.5 grams.
To solve this problem, we need to calculate the probability using the Z-Score formula for sample means. The Z-Score formula is given by:
Z = (X - μ) / (σ / √n)
Where:
Z is the Z-Score
X is the sample mean
μ is the population mean
σ is the population standard deviation
n is the sample size
Given:
Sample mean (X) = 402.7 grams
Population standard deviation (σ) = 8.8 grams
Sample size (n) = 40
Step 1: Calculate the Z-Score
Z = (X - μ) / (σ / √n)
= (402.7 - μ) / (8.8 / √40)
Step 2: Look up the Z-Score in the Z-table
The Z-table provides the probability corresponding to a given Z-Score. In this case, we want to find the probability that the mean weight exceeds 405.5 grams, which means we need to calculate the area to the right of the Z-Score.
Step 3: Calculate the probability
Using the Z-table, find the probability corresponding to the Z-Score calculated in step 1. Subtract this probability from 1 to obtain the probability that the mean weight exceeds 405.5 grams.
Note: Since the population standard deviation is known, we can use the Z-Score formula instead of the t-distribution.
I will calculate the Z-Score and look it up in the Z-table.