To determine the minimum sample size needed for the scientist to be 90% confident that her estimate is within 0.6 pounds of the mean birth weight, we need to use the formula for the sample size $n$ in estimating a population mean with a specified margin of error $E$:

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

where:

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

Given:

• Confidence level = 90%
• Population standard deviation $\sigma = 2.5$ pounds
• Margin of error $E = 0.6$ pounds

First, we find the z-value for a 90% confidence level. The z-value corresponds to the critical value that leaves 5% in each tail of the standard normal distribution (since 90% confidence level leaves 10% total in the two tails, 5% in each). We can look up this value in a standard normal distribution table or use a calculator.

The z-value for 90% confidence is approximately 1.645.

Now, we can plug in the values into the sample size formula:

$n = \left( \frac{1.645 \cdot 2.5}{0.6} \right)^2$

Calculating the intermediate steps:

$\frac{1.645 \cdot 2.5}{0.6} = \frac{4.1125}{0.6} \approx 6.854$

Next, we square this value:

$(6.854)^2 \approx 47.001$

Since the sample size $n$ must be a whole number, we round up to the next whole number to ensure the margin of error requirement is met:

$n \approx 48$

Therefore, the minimum sample size needed is:

$\boxed{48}$

Would you like details or have any questions?

Here are 8 related questions that expand on this information:

1. What is the margin of error if the sample size is increased to 100?
2. How does changing the confidence level to 95% affect the required sample size?
3. What is the impact of a larger standard deviation on the required sample size?
4. Can you explain the concept of a confidence interval in the context of this problem?
5. How is the z-value determined for different confidence levels?
6. What assumptions are made in this sample size calculation?
7. How would you calculate the sample size if the standard deviation was not known?
8. What is the significance of using a random sample in this study?

Tip: Always round up the sample size to ensure the desired margin of error is not exceeded.