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Let’s assume that 24% of all M&M’s are blue. This means the population proportion of M&M’s that are blue is p = 0.24. Suppose random samples of 25 M&M’s are obtained and the proportion of blue M&M’s are found. The image below displays a simulated distribution of 300 sample proportions of M&M’s that are blue, for samples of size n = 25. Estimate the mean of the sample proportions from the dotplot.

The given dot plot represents the simulated distribution of 300 sample proportions of blue M&M's, where the sample size \( n = 25 \) and the population proportion \( p = 0.24 \).

From the dot plot, we can estimate the mean of the sample proportions based on the peak and central tendency. The peak appears at \( 0.24 \), which aligns with the population proportion \( p = 0.24 \). Given the symmetry of the distribution and that it centers around 0.24, it is reasonable to estimate the mean of the sample proportions as approximately \( 0.24 \).

Thus, the mean of the sample proportions is estimated to be:

\[
\hat{\mu}_{\hat{p}} \approx 0.24
\]

Would you like more details on how this distribution relates to the population proportion, or how sampling variability impacts the sample mean?

### Related Questions:
1. What is the expected standard deviation of the sample proportions in this case?
2. How does the Central Limit Theorem apply to this distribution?
3. How would increasing the sample size \( n \) affect the shape of the distribution?
4. What does it mean if a sample proportion in this simulation is far from the population proportion?
5. How can you calculate the standard error for the sample proportions?

### Tip:
The mean of sample proportions in a random sampling simulation will always approximate the population proportion, as long as the sampling method is unbiased.

Learn how to estimate the mean of sample proportions for blue M&M's using a simulated distribution of 300 samples with a sample size of n = 25. This problem applies key statistical concepts like sampling distributions and the Central Limit Theorem. Ideal for college-level statistics students.

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