To answer these questions, we'll apply the 68-95-99.7 rule (empirical rule) for normal distributions. This rule states that:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

Given Information:

• Mean ($\mu$) = 40
• Standard Deviation ($\sigma$) = 5

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1. Percentage of observations between 35 and 45:

The range from 35 to 45 corresponds to 1 standard deviation below and above the mean.

• 35 is $40 - 5$ (1 standard deviation below the mean).
• 45 is $40 + 5$ (1 standard deviation above the mean).

Using the empirical rule, 68% of the observations lie between 35 and 45.

So, the percentage of observations between 35 and 45 is:

• 68%

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2. Percentage of observations between 30 and 50:

The range from 30 to 50 corresponds to 2 standard deviations below and above the mean.

• 30 is $40 - 2(5)$ (2 standard deviations below the mean).
• 50 is $40 + 2(5)$ (2 standard deviations above the mean).

Using the empirical rule, 95% of the observations lie between 30 and 50.

So, the percentage of observations between 30 and 50 is:

• 95%

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3. Percentage of observations less than 35:

The value 35 is 1 standard deviation below the mean.

• Since 68% of the data is within 1 standard deviation, we know that half of that (34%) is below the mean (40).
• Therefore, the percentage of observations less than 35 would be half of 68% (34%), plus the additional 34% below the mean, which makes it:
  • 34% (below the mean) + 34% (half of the 68%) = 34%

Thus, the percentage of observations less than 35 is approximately:

• 34.0%

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Summary of answers:

1. 68% of the observations fall between 35 and 45.
2. 95% of the observations fall between 30 and 50.
3. 34.0% of the observations are less than 35.

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Would you like further clarification or a breakdown of any of these calculations? Also, here are some follow-up questions:

1. What is the percentage of observations between 40 and 45?
2. How would you calculate the percentage of observations above 45?
3. How do we calculate the percentage of observations between 30 and 40?
4. What are the advantages of using the empirical rule for normal distributions?
5. What happens if the distribution is not normal—can the empirical rule still be applied?

Tip: Remember that the empirical rule is most accurate for symmetric, bell-shaped (normal) distributions.