Question

The following table presents the time taken to review articles that were submitted for publication to a particular journal during a recent year. A few articles took longer than 6 months to review; these are omitted from the table. Consider the data to be a population.
Time: 0-0.9 with the number of articles as 44
1.0-1.9 with the number of articles as 16
2.0-2.9 with the number of articles as 17
3.0-3.9 with the number of articles as 20
4.0-4.9 with the number of articles as 12
5.0-5.9 with the number of articles as 15

(a) Approximate the variance of the times. Round the answer to at least two decimal places.

The variance is
.

(b) Approximate the standard deviation of the times. Round the answer to at least two decimal places.

The standard deviation is
.

Answer 1

B is correct but A is not

Answer 2

To calculate the variance and standard deviation of the given data, we can utilize the method of grouped data.

### Step 1: Determine Midpoints (x)

First, we need to calculate the midpoint of each time interval. The midpoint is calculated as the average of the lower and upper bounds of each interval.

| Time Interval | Midpoint (x) | Frequency (f) |
|---------------|--------------|---------------|
| 0.0 - 0.9 | \(0.45\) | \(44\) |
| 1.0 - 1.9 | \(1.45\) | \(16\) |
| 2.0 - 2.9 | \(2.45\) | \(17\) |
| 3.0 - 3.9 | \(3.45\) | \(20\) |
| 4.0 - 4.9 | \(4.45\) | \(12\) |
| 5.0 - 5.9 | \(5.45\) | \(15\) |

### Step 2: Calculate the Mean (μ)

The mean (\(\mu\)) of the population can be calculated using the formula:

\[
\mu = \frac{\sum f \cdot x}{\sum f}
\]

Calculating \(\sum f \cdot x\) and \(\sum f\):

- \( \sum f \cdot x = (44 \times 0.45) + (16 \times 1.45) + (17 \times 2.45) + (20 \times 3.45) + (12 \times 4.45) + (15 \times 5.45) \)
- \( \sum f = 44 + 16 + 17 + 20 + 12 + 15 = 124 \)

Calculating \( \sum f \cdot x \):

\[
\sum f \cdot x = 19.8 + 23.2 + 41.65 + 69 + 53.4 + 81.75 = 288.9
\]

Now, calculate the mean:

\[
\mu = \frac{288.9}{124} \approx 2.33
\]

### Step 3: Calculate the Variance (σ²)

Using the formula for variance:

\[
\sigma^2 = \frac{\sum f \cdot (x - \mu)^2}{\sum f}
\]

First, compute \( (x - \mu)^2 \) for each midpoint:

- For \(0.45\): \( (0.45 - 2.33)^2 = 3.5049 \)
- For \(1.45\): \( (1.45 - 2.33)^2 = 0.7744 \)
- For \(2.45\): \( (2.45 - 2.33)^2 = 0.0144 \)
- For \(3.45\): \( (3.45 - 2.33)^2 = 1.2549 \)
- For \(4.45\): \( (4.45 - 2.33)^2 = 4.5296 \)
- For \(5.45\): \( (5.45 - 2.33)^2 = 9.1849 \)

Now we multiply by frequency (\(f\)) and sum this:

\[
\sum f \cdot (x - \mu)^2 = (44 \cdot 3.5049) + (16 \cdot 0.7744) + (17 \cdot 0.0144) + (20 \cdot 1.2549) + (12 \cdot 4.5296) + (15 \cdot 9.1849
\]

Calculating:

\[
= 154.2206 + 12.3904 + 0.2448 + 25.098 + 54.3552 + 137.7235 = 383.0225
\]

Now calculate the variance:

\[
\sigma^2 = \frac{383.0225}{124} \approx 3.08
\]

### Step 4: Calculate the Standard Deviation (σ)

The standard deviation can be calculated as:

\[
\sigma = \sqrt{\sigma^2}
\]

Calculating:

\[
\sigma \approx \sqrt{3.08} \approx 1.75
\]

### Final Answers

(a) The variance is **3.08**.

(b) The standard deviation is **1.75**.

Answer 3

Let's revisit the calculations for the variance step-by-step to ensure we get the correct result.

### Step 1: Calculate the Mean (μ)

From the previous response, we found:

- \(\sum f = 124\)
- \(\sum f \cdot x = 288.9\)

The mean (\(\mu\)) was calculated as:

\[
\mu = \frac{288.9}{124} \approx 2.33
\]

### Step 2: Calculate \( (x - \mu)^2 \) values

The midpoints \(x\) and corresponding \( (x - \mu)^2 \) calculations are as follows:

1. **For 0.45**:
\[
(0.45 - 2.33)^2 \approx (1.88)^2 = 3.5344
\]
2. **For 1.45**:
\[
(1.45 - 2.33)^2 \approx (0.88)^2 = 0.7744
\]
3. **For 2.45**:
\[
(2.45 - 2.33)^2 \approx (0.12)^2 = 0.0144
\]
4. **For 3.45**:
\[
(3.45 - 2.33)^2 \approx (1.12)^2 = 1.2544
\]
5. **For 4.45**:
\[
(4.45 - 2.33)^2 \approx (2.12)^2 = 4.4944
\]
6. **For 5.45**:
\[
(5.45 - 2.33)^2 \approx (3.12)^2 = 9.7344
\]

### Step 3: Compute \(f \cdot (x - \mu)^2\)

Now let's compute \(f \cdot (x - \mu)^2\):

1. **For 0.45**:
\[
44 \cdot 3.5344 \approx 155.8016
\]
2. **For 1.45**:
\[
16 \cdot 0.7744 \approx 12.3904
\]
3. **For 2.45**:
\[
17 \cdot 0.0144 \approx 0.2448
\]
4. **For 3.45**:
\[
20 \cdot 1.2544 \approx 25.088
\]
5. **For 4.45**:
\[
12 \cdot 4.4944 \approx 53.9328
\]
6. **For 5.45**:
\[
15 \cdot 9.7344 \approx 145.016
\]

### Step 4: Calculate \(\sum f \cdot (x - \mu)^2\)

\[
\sum f \cdot (x - \mu)^2 \approx 155.8016 + 12.3904 + 0.2448 + 25.088 + 53.9328 + 145.016 \approx 392.474
\]

### Step 5: Calculate the Variance (\(\sigma^2\))

Now, substituting back into the variance formula:

\[
\sigma^2 = \frac{\sum f \cdot (x - \mu)^2}{\sum f} = \frac{392.474}{124} \approx 3.17
\]

### Step 6: Standard Deviation (\(\sigma\))

Now, calculating the standard deviation:

\[
\sigma = \sqrt{3.17} \approx 1.78
\]

### Final Results

(a) The corrected variance is **3.17**.

(b) The standard deviation is **1.78**.

Thank you for your patience! Please let me know if you need further clarification or assistance!