To calculate the required sample size, we use the formula for the sample size $n$ when estimating a population mean:

$n = \left( \frac{Z \cdot \sigma}{E} \right)^2$

Where:

• $Z$ is the Z-score corresponding to the desired confidence level.
• $\sigma$ is the population standard deviation.
• $E$ is the margin of error.

Step 1: Determine the Z-score

For a 95% confidence level, the Z-score is approximately $1.96$.

Step 2: Plug in the values

• $\sigma = 115$ (standard deviation)
• $E = 40$ (margin of error)
• $Z = 1.96$

Substitute these into the formula:

$n = \left( \frac{1.96 \cdot 115}{40} \right)^2$

Step 3: Simplify the calculation

1. Calculate $\frac{1.96 \cdot 115}{40}$: $\frac{1.96 \cdot 115}{40} = \frac{225.4}{40} = 5.635$

2. Square this value: $n = (5.635)^2 = 31.75$

Step 4: Round up

Since sample size must be an integer, we round up to the nearest whole number:

$n = 32$

Conclusion

The required sample size to estimate the mean weight of BC bears to within 40 pounds with 95% confidence is 32.

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Would you like further details or have any questions? Here are 5 related questions to expand on this:

1. How does changing the confidence level affect the required sample size?
2. What happens to the required sample size if the margin of error is reduced to 20 pounds?
3. Can this formula be applied if the population standard deviation is unknown?
4. Why do we always round up the sample size to the nearest whole number?
5. How is this method different from calculating sample size for proportions?

Tip: Always round up when calculating sample sizes to ensure the desired precision is achieved.