The correct answer is:

Yes, the sample values were randomly selected and the distribution of sample means for each population is normal because a histogram of each sample is approximately normal.

Explanation:

• Random Selection: The problem clearly states that the employees were randomly assigned to either the electric bicycle or electric scooter groups. This satisfies the requirement for random sampling.

• Normality: The histograms of both sample groups (bicycles and scooters) show that the data is approximately normal, as stated in the problem. For a hypothesis test or confidence interval for the difference in means, the sample distributions should ideally be normal or at least close to normal. The visual representation in the histograms suggests that both distributions are roughly symmetric, which is a good indicator of normality.

• Sample Size: Although the sample size for each group is not particularly large (13 for bicycles and 11 for scooters), it is generally accepted that if the sample sizes are greater than 30, the Central Limit Theorem would apply, even if the original distributions are not normal. Since the sample sizes here are smaller, we rely more on the histogram shape to determine normality. The histograms suggest that the data is not highly skewed, so it should be reasonable to proceed with inference about the means.

The other options are incorrect because:

• Option 1: The samples were indeed randomly selected, so this part of the statement is incorrect.
• Option 2: The histograms do show that the distributions of both sample groups are approximately normal, so this is not an issue here.
• Option 3: The histograms do suggest that the sample distributions are approximately normal.
• Option 4: This option talks about the Central Limit Theorem, but since we have histograms showing approximate normality, we don't need to rely on the Central Limit Theorem for smaller sample sizes here.

To recap:

• Randomness in selection is confirmed.
• Histograms suggest the distributions are roughly normal.
• The sample sizes, though small, do not hinder the ability to proceed with the analysis based on the histogram shape.

Let me know if you need further clarification or a deeper dive into any of these points!

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Here are some follow-up questions to explore related concepts:

1. How does the Central Limit Theorem apply when the sample size is small but the data is approximately normal?
2. What assumptions should be checked before using a t-test for comparing means?
3. How does one determine if a sample distribution is normal without a histogram (e.g., using a Q-Q plot)?
4. Why is random sampling so important in statistical inference?
5. How does sample size affect the power of a hypothesis test?

Tip: When you're unsure about the normality of data, always plot it! Histograms or Q-Q plots give a good visual cue about the underlying distribution.