Answer :
Final answer:
The smallest sample size necessary to estimate the mean weight of all koalas in the region with a 90% confidence level and a margin of error of +0.5 pounds is approximately 252,250.
Explanation:
To determine the smallest sample size necessary to estimate the mean weight of all koalas in a certain region, we can use the formula for sample size calculation in a normal distribution:
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 margin of error
In this case, the researcher wants to be 90% confident that her sample estimate is accurate to within +0.5 pounds of the population mean. The population standard deviation is given as 305 pounds.
Substituting the given values into the formula:
n = (Z * σ / E)^2
Plugging in the values:
n = (Z * 305 / 0.5)^2
Since we want to be 90% confident, the Z-score corresponding to a 90% confidence level is approximately 1.645.
n = (1.645 * 305 / 0.5)^2
Calculating the sample size:
n = (502.225)^2
n ≈ 252,250
Therefore, the smallest sample size necessary to estimate the mean weight of all koalas in the region with a 90% confidence level and a margin of error of +0.5 pounds is approximately 252,250.
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