Question 9 A student visits a population of five local restaurants $$ (\mathrm{N}=5) $$ and rates each as follows: $$ 4,3,6,4 $$, and 8 , Using the computational formula, what is the SS and $$ \sigma^{2} $$ ? (A) $$ 17 ; 3.4 $$ (B) $$ 20 ; 4 $$ (C) $$ 18 ; 3.6 $$ (D) $$ 19 ; 3.8 $$

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To find the sum of squares (SS) and the variance ($$\sigma^2$$) of the ratings given by the student, we can follow these steps:

1. **Calculate the mean ($$\bar{x}$$) of the ratings.**
2. **Calculate the sum of squares (SS).**
3. **Calculate the variance ($$\sigma^2$$).**

### Step 1: Calculate the Mean

The ratings are: $$4, 3, 6, 4, 8$$.

The mean is calculated as follows:

$$
\bar{x} = \frac{\sum x_i}{N}
$$

Where $$N$$ is the number of ratings.

### Step 2: Calculate the Sum of Squares (SS)

The sum of squares is calculated using the formula:

$$
SS = \sum (x_i - \bar{x})^2
$$

### Step 3: Calculate the Variance ($$\sigma^2$$)

The variance is calculated as:

$$
\sigma^2 = \frac{SS}{N}
$$

Now, let's perform the calculations step by step.

1. Calculate the mean:

$$
\bar{x} = \frac{4 + 3 + 6 + 4 + 8}{5} = \frac{25}{5} = 5
$$

2. Calculate the sum of squares (SS):

$$
SS = (4 - 5)^2 + (3 - 5)^2 + (6 - 5)^2 + (4 - 5)^2 + (8 - 5)^2
$$

Calculating each term:

- $$(4 - 5)^2 = 1$$
- $$(3 - 5)^2 = 4$$
- $$(6 - 5)^2 = 1$$
- $$(4 - 5)^2 = 1$$
- $$(8 - 5)^2 = 9$$

Now summing these:

$$
SS = 1 + 4 + 1 + 1 + 9 = 16
$$

3. Calculate the variance ($$\sigma^2$$):

$$
\sigma^2 = \frac{SS}{N} = \frac{16}{5} = 3.2
$$

Now, let's summarize the results:

- Sum of Squares (SS) = 16
- Variance ($$\sigma^2$$) = 3.2

However, it seems I made a mistake in the calculations. Let's recalculate the SS and variance correctly.

### Recalculating SS and Variance

1. **Mean**:

$$
\bar{x} = 5
$$

2. **Sum of Squares (SS)**:

$$
SS = (4 - 5)^2 + (3 - 5)^2 + (6 - 5)^2 + (4 - 5)^2 + (8 - 5)^2
$$

Calculating each term again:

- $$(4 - 5)^2 = 1$$
- $$(3 - 5)^2 = 4$$
- $$(6 - 5)^2 = 1$$
- $$(4 - 5)^2 = 1$$
- $$(8 - 5)^2 = 9$$

Now summing these:

$$
SS = 1 + 4 + 1 + 1 + 9 = 16
$$

3. **Variance ($$\sigma^2$$)**:

$$
\sigma^2 = \frac{SS}{N} = \frac{16}{5} = 3.2
$$

### Final Results

- SS = 16
- Variance ($$\sigma^2$$) = 3.2

Since the options provided do not match the calculated values, let's check the calculations again for any errors.

### Correcting the Calculation

Let's calculate the SS and variance using the computational formula:

1. **Calculate the mean**:

$$
\bar{x} = \frac{4 + 3 + 6 + 4 + 8}{5} = 5
$$

2. **Calculate SS using the formula**:

$$
SS = \sum x_i^2 - N \cdot \bar{x}^2
$$

Calculating $$\sum x_i^2$$:

$$
\sum x_i^2 = 4^2 + 3^2 + 6^2 + 4^2 + 8^2 = 16 + 9 + 36 + 16 + 64 = 141
$$

Now substituting into the SS formula:

$$
SS = 141 - 5 \cdot 5^2 = 141 - 125 = 16
$$

3. **Calculate variance**:

$$
\sigma^2 = \frac{SS}{N} = \frac{16}{5} = 3.2
$$

### Conclusion

The calculations confirm that the SS is 16 and the variance is 3.2. However, since the options provided do not match, it seems there might be a misunderstanding in the problem or the options given.

Please verify the problem statement or the options provided.
The sum of squares (SS) is 16 and the variance ($$\sigma^2$$) is 3.2.

Question 9 A student visits a population of five local restaurants \( (\mathrm{N}=5) \) and rates each as follows: \( 4,3,6,4 \), and 8 , Using the computational formula, what is the SS and \( \sigma^{2} \) ? (A) \( 17 ; 3.4 \) (B) \( 20 ; 4 \) (C) \( 18 ; 3.6 \) (D) \( 19 ; 3.8 \)

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