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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.

Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The executives hire a statistical consultant and ask her to determine the mean shopping time, μ, of customers at the supermarkets. The consultant will collect a random sample of shopping times at the supermarkets and use the mean of these shopping times to estimate μ. Assuming that the standard deviation of the population of shopping times at the supermarkets is 29 minutes, what is the minimum sample size she must collect in order for her to be 95% confident that her estimate is within 3 minutes of μ?

Learn how to calculate the minimum sample size required to estimate the mean shopping time with 95% confidence. This problem involves the application of sample size formulas, confidence intervals, and the normal distribution, using a population standard deviation of 29 minutes and a margin of error of 3 minutes.

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