The frequency distribution was obtained using a class width of 0.5 for data on cigarette tax rates. Use the frequency distribution to approximate the population mean and population standard deviation. Compare these results to the actual mean muequals​1.608 and standard deviation sigmaequals​1.065.

Tax Rate	Lower Limit	Upper Limit	Frequency
0.00-0.49	0.00	0.49	6
0.50-0.99	0.50	0.99	13
1.00-1.49	1.00	1.49	6
1.50-1.99	1.50	1.99	6
2.00-2.49	2.00	2.49	5
2.50-2.99	2.50	2.99	5
3.00-3.49	3.00	3.49	3
3.50-3.99	3.50	3.99	3
4.00-4.49	4.00	4.49	1

Whats the population mean

To approximate the population mean using the given frequency distribution, we can apply the **midpoint method**. Here's the step-by-step process:

### Step 1: Calculate the midpoint for each class
The midpoint \((x_i)\) of each class is calculated by averaging the lower and upper limits of the class.

\[
\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}
\]

### Step 2: Multiply each midpoint by its corresponding frequency to get the class total contribution
The total contribution from each class is the product of the midpoint and the frequency.

\[
\text{Class contribution} = x_i \times f_i
\]

### Step 3: Compute the approximate population mean using the formula:

\[
\bar{x} = \frac{\sum (x_i \times f_i)}{\sum f_i}
\]

Where:
- \(x_i\) is the midpoint of the \(i\)-th class.
- \(f_i\) is the frequency of the \(i\)-th class.
- \(\sum f_i\) is the total frequency (total number of data points).

Now, let's calculate.

| Tax Rate | Midpoint \(x_i\) | Frequency \(f_i\) | \(x_i \times f_i\) |
|----------|------------------|-------------------|--------------------|
| 0.00-0.49 | 0.245            | 6                 | 1.47               |
| 0.50-0.99 | 0.745            | 13                | 9.685              |
| 1.00-1.49 | 1.245            | 6                 | 7.47               |
| 1.50-1.99 | 1.745            | 6                 | 10.47              |
| 2.00-2.49 | 2.245            | 5                 | 11.225             |
| 2.50-2.99 | 2.745            | 5                 | 13.725             |
| 3.00-3.49 | 3.245            | 3                 | 9.735              |
| 3.50-3.99 | 3.745            | 3                 | 11.235             |
| 4.00-4.49 | 4.245            | 1                 | 4.245              |

### Step 4: Sum the class contributions and the frequencies

- \(\sum (x_i \times f_i) = 79.26\)
- \(\sum f_i = 48\)

### Step 5: Calculate the approximate population mean

\[
\bar{x} = \frac{79.26}{48} \approx 1.651
\]

### Comparison with the actual mean:

- The actual mean: \(\mu = 1.608\)
- Approximate mean: \(\bar{x} \approx 1.651\)

So, the approximate population mean is **1.651**, which is very close to the actual mean of 1.608.

Would you like me to compute the standard deviation as well or explain anything further?

---

### Related Questions:
1. How do we calculate the standard deviation from a frequency distribution?
2. What is the difference between sample mean and population mean?
3. How does class width affect the accuracy of approximations in grouped data?
4. What assumptions do we make when calculating means from frequency distributions?
5. Can we use the midpoint method in situations other than frequency distributions?

### Tip:
When working with grouped data, the accuracy of approximations improves with smaller class widths, as the midpoint becomes a better representative of individual data points within each class.

The frequency distribution was obtained using a class width of 0.5 for data on cigarette tax rates. Use the frequency distribution to approximate the population mean and population standard deviation. Compare these results to the actual mean muequals​1.608 and standard deviation sigmaequals​1.065.

Learn how to approximate the population mean and standard deviation using a frequency distribution of cigarette tax rates. This problem covers essential statistical concepts such as the midpoint method and grouped data. Compare your results with the actual mean of 1.608 and standard deviation of 1.065. Suitable for Grades 10-12.

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