Final answer:
To estimate the mean weight of mice to within 0.5 grams with 80% confidence, we use the sample size formula involving a z-value, the standard deviation, and margin of error. The minimum required sample size calculation yields 106 mice.
Step-by-step explanation:
To estimate the mean weight of mice after being fed a special diet to within 0.5 grams with 80% confidence, the minimum required sample size is needed. This can be calculated using the formula for sample size in hypothesis testing or confidence interval calculations:
n = (Z * σ / E)^2
Where:
- n is the sample size
- Z is the z-value corresponding to the desired confidence level
- σ (sigma) is the population standard deviation
- E is the margin of error
For an 80% confidence interval, the z-value (Z) is approximately 1.282. The known standard deviation (σ) is 4 grams, and the desired margin of error (E) is 0.5 grams.
Let's plug the values into the formula:
n = (1.282 * 4 / 0.5)^2
n = (5.128 / 0.5)^2
n = (10.256)^2
n = 105.12
Since we cannot have a fraction of a mouse, we'll round up to the nearest whole number. Therefore, the minimum required sample size is 106 mice.