Question
A lake has 1,200 fish whose lengths (in centimeters) are normally distributed with:
a mean length, in cm, of 30
a standard deviation, in cm, of 4 +1
Estimate how many fish are shorter than 25 cm.
Click Here for a Hint
(2 points)
Responses
75 fish
75 fish
127 fish
127 fish
190 fish
190 fish
300 fish
a mean length, in cm, of 30
a standard deviation, in cm, of 4 +1
Estimate how many fish are shorter than 25 cm.
Click Here for a Hint
(2 points)
Responses
75 fish
75 fish
127 fish
127 fish
190 fish
190 fish
300 fish
Answers
GPT-4o mini
To estimate how many fish are shorter than 25 cm, we can use the properties of the normal distribution.
1. We first need to calculate the z-score for 25 cm using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where \(X\) is the value we are interested in (25 cm), \(\mu\) is the mean (30 cm), and \(\sigma\) is the standard deviation (4 cm).
Substituting in the values:
\[
z = \frac{(25 - 30)}{4} = \frac{-5}{4} = -1.25
\]
2. Next, we look up the z-score of -1.25 in the standard normal distribution table, or use a calculator that provides the cumulative distribution function (CDF). The z-score table gives us the proportion of the distribution that is less than a given z-score.
The cumulative probability (proportion of fish shorter than 25 cm) for \(z = -1.25\) is approximately 0.1056.
3. To find the estimated number of fish shorter than 25 cm, we multiply the total number of fish by this proportion:
\[
\text{Number of fish shorter than 25 cm} = 1200 \times 0.1056 \approx 126.72
\]
Since we can't have a fraction of a fish, we round this to about 127 fish.
Therefore, the estimate is **127 fish**.
1. We first need to calculate the z-score for 25 cm using the formula:
\[
z = \frac{(X - \mu)}{\sigma}
\]
where \(X\) is the value we are interested in (25 cm), \(\mu\) is the mean (30 cm), and \(\sigma\) is the standard deviation (4 cm).
Substituting in the values:
\[
z = \frac{(25 - 30)}{4} = \frac{-5}{4} = -1.25
\]
2. Next, we look up the z-score of -1.25 in the standard normal distribution table, or use a calculator that provides the cumulative distribution function (CDF). The z-score table gives us the proportion of the distribution that is less than a given z-score.
The cumulative probability (proportion of fish shorter than 25 cm) for \(z = -1.25\) is approximately 0.1056.
3. To find the estimated number of fish shorter than 25 cm, we multiply the total number of fish by this proportion:
\[
\text{Number of fish shorter than 25 cm} = 1200 \times 0.1056 \approx 126.72
\]
Since we can't have a fraction of a fish, we round this to about 127 fish.
Therefore, the estimate is **127 fish**.