The problem in the image is asking for the minimum sample size needed for a statistical estimate. The given details are:

• The population standard deviation (σ) is 29 minutes.
• The desired margin of error (E) is 3 minutes.
• The confidence level is 95%, which corresponds to a Z-score of 1.96 (from the standard normal distribution).

The formula for determining the minimum sample size $n$ for estimating the mean is:

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

Where:

• $Z$ is the Z-score (1.96 for a 95% confidence level),
• $\sigma$ is the population standard deviation (29 minutes),
• $E$ is the margin of error (3 minutes).

Let's calculate this now.The minimum sample size required is 359 to ensure the estimate of the mean shopping time is within 3 minutes of the true mean with 95% confidence.

Do you want further details or have any questions?

Here are 5 related questions to expand on this concept:

1. What happens to the required sample size if we reduce the margin of error to 2 minutes?
2. How does increasing the confidence level to 99% affect the sample size?
3. Why do we use a Z-score of 1.96 for a 95% confidence level?
4. Can the sample size formula be used if the population standard deviation is unknown?
5. How would the sample size change if the population standard deviation were 35 minutes instead of 29?

Tip: Always round up when calculating sample size, as you can't collect a fraction of a participant.