The weights of trout in a certain river follow a normal distribution with mean 3 pounds and standard deviation 0.8 pounds. What proportion of trout weigh between 3.2 pounds and 4.8 pounds?
a.) .3891
b.) .9972
c.) .9452
d.) .5865
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
d. 5865
To find the proportion of trout that weigh between 3.2 pounds and 4.8 pounds, we need to calculate the z-scores for these two weights and then look up the corresponding probabilities from the standard normal distribution table.
First, let's calculate the z-score for 3.2 pounds using the formula:
z = (x - μ) / σ
where x is the given weight, μ is the mean, and σ is the standard deviation.
Using the given values, we have:
z1 = (3.2 - 3) / 0.8
z1 = 0.2 / 0.8
z1 = 0.25
Next, let's calculate the z-score for 4.8 pounds:
z2 = (4.8 - 3) / 0.8
z2 = 1.8 / 0.8
z2 = 2.25
Now, we need to find the probabilities associated with these z-scores from the standard normal distribution table. The proportion between these two weights is given by the difference in these probabilities.
Using the standard normal distribution table or a calculator, we find that the probability corresponding to z1 = 0.25 is approximately 0.5987, and the probability corresponding to z2 = 2.25 is approximately 0.9878.
Therefore, the proportion of trout that weigh between 3.2 pounds and 4.8 pounds is:
0.9878 - 0.5987 = 0.3891
So, the correct answer is a.) 0.3891.
To find the proportion of trout that weigh between 3.2 pounds and 4.8 pounds, we need to calculate the area under the normal distribution curve between these two weights.
Step 1: Standardize the weights:
We first need to standardize the weights using the formula: z = (X - μ) / σ, where X is the specific weight, μ is the mean, and σ is the standard deviation.
For the lower bound of 3.2 pounds:
z1 = (3.2 - 3) / 0.8 = 0.2 / 0.8 = 0.25
For the upper bound of 4.8 pounds:
z2 = (4.8 - 3) / 0.8 = 1.8 / 0.8 = 2.25
Step 2: Find the corresponding proportions:
Next, we need to find the proportions of the area under the normal distribution curve between the standardized values z1 and z2. We can use a standard normal distribution table or calculator to do this.
From the standard normal distribution table, we look up the value of 0.25, which is 0.5987.
Similarly, we look up the value of 2.25, which is 0.9878.
Step 3: Calculate the proportion:
To find the proportion of trout weighing between 3.2 pounds and 4.8 pounds, we subtract the lower value from the upper value:
Proportion = 0.9878 - 0.5987 = 0.3891
Therefore, the proportion of trout weighing between 3.2 pounds and 4.8 pounds is approximately 0.3891.
Thus, the correct answer is a.) 0.3891.