To solve this problem, we need to determine how many data points fall within two population standard deviations of the mean for the given data set. Here's the step-by-step breakdown:

1. Calculate the Mean ($\mu$)

The mean is the sum of the data values multiplied by their respective frequencies, divided by the total frequency (number of data points).

$\mu = \frac{\sum (x \cdot f)}{\sum f}$ Where $x$ represents each data value, and $f$ represents its frequency.

2. Calculate the Population Standard Deviation ($\sigma$)

The population standard deviation formula is:

$\sigma = \sqrt{\frac{\sum f(x - \mu)^2}{\sum f}}$

3. Determine the Interval for "Within 2 Standard Deviations"

Once we have the mean and standard deviation, we calculate the range of values that fall within two standard deviations of the mean:

$\text{Lower Bound} = \mu - 2\sigma$ $\text{Upper Bound} = \mu + 2\sigma$

4. Count the Data Points in This Interval

Finally, we count how many data points (including their frequencies) fall within the calculated range.

Let me perform the calculations.### Results:

1. Mean ($\mu$): $33.23$
2. Population Standard Deviation ($\sigma$): $4.75$
3. Lower Bound (within 2 standard deviations): $23.73$
4. Upper Bound (within 2 standard deviations): $42.74$
5. Number of Data Points within 2 Standard Deviations: 84 data points

Thus, 84 data points fall within two standard deviations of the mean.

Would you like further details or explanations?

Relative Questions:

1. How do you calculate the mean and standard deviation for grouped data?
2. What is the difference between population and sample standard deviation?
3. How can you visualize the distribution of data within standard deviations?
4. Why is the "2 standard deviations" rule important in statistics?
5. How would the calculation change if the data were normally distributed?

Tip: When working with frequency distributions, always ensure that you're summing both the data values and frequencies correctly for accurate results.