A study of elephants wishes to determine the average weight of a certain subspecies of elephants. The standard deviation of the population is 2,014 pounds. How many elephants need to be weighed so we can be 93% confident to be accurate within 325 pounds?
To determine the sample size needed to estimate the average weight of a certain subspecies of elephants with a desired level of confidence and a specified margin of error, we can use the formula for sample size calculation:
n = [(Z * σ) / E]^2
Where:
- n is the sample size
- Z is the z-score corresponding to the desired level of confidence
- σ is the population standard deviation
- E is the specified margin of error
In this case, we want to be 93% confident and accurate within 325 pounds for the average weight of the subspecies of elephants. First, let's find the z-score corresponding to a 93% confidence level.
Step 1: Finding the Z-Score
To find the z-score, we need to determine the area to the right of the z-score on a standard normal distribution table.
Area to the right = (1 - Confidence Level) / 2
= (1 - 0.93) / 2
= 0.07 / 2
= 0.035
Using a standard normal distribution table or a statistical calculator, we find that the z-score corresponding to an area of 0.035 to the right is approximately 1.812.
Step 2: Calculating the Sample Size
Now, we can substitute the values into the sample size formula:
n = [(Z * σ) / E]^2
= [(1.812 * 2014) / 325]^2
Calculating this formula, we get:
n ≈ 123 elephants
Therefore, to be 93% confident and accurate within 325 pounds for the average weight of the subspecies of elephants, we would need to weigh approximately 123 elephants.