Question

twenty - five randomly selected students were asked the number of movies they watched the previous week. the results are as follows.

# of movies	frequency
1	9
2	7
3	3
4	2

(a) find the sample mean \\(\bar{x}\\). (enter your answer to two decimal places.)
\\(\bar{x}=\\)
(b) find the approximate sample standard deviation, \\(s\\). (round your answer to two decimal places.)
\\(s=\\)

Explanation:

Part (a): Finding the Sample Mean $\bar{x}$

Step 1: Calculate the product of each number of movies and its frequency

For each row in the table, multiply the number of movies ($x$) by its frequency ($f$).

• For $x = 0$, $f = 4$: $0 \times 4 = 0$
• For $x = 1$, $f = 9$: $1 \times 9 = 9$
• For $x = 2$, $f = 7$: $2 \times 7 = 14$
• For $x = 3$, $f = 3$: $3 \times 3 = 9$
• For $x = 4$, $f = 2$: $4 \times 2 = 8$

Step 2: Sum the products and the frequencies

First, sum the products: $0 + 9 + 14 + 9 + 8 = 40$
Then, sum the frequencies: $4 + 9 + 7 + 3 + 2 = 25$ (which matches the number of students)

Step 3: Calculate the sample mean

The formula for the sample mean with frequency is $\bar{x} = \frac{\sum (x \times f)}{\sum f}$.
Substitute the sums: $\bar{x} = \frac{40}{25} = 1.60$

Part (b): Finding the Sample Standard Deviation $s$

Step 1: Calculate the squared deviations from the mean, multiplied by frequency

First, recall the mean $\bar{x} = 1.60$. For each $x$:

• For $x = 0$: $(0 - 1.60)^2 \times 4 = (-1.60)^2 \times 4 = 2.56 \times 4 = 10.24$
• For $x = 1$: $(1 - 1.60)^2 \times 9 = (-0.60)^2 \times 9 = 0.36 \times 9 = 3.24$
• For $x = 2$: $(2 - 1.60)^2 \times 7 = (0.40)^2 \times 7 = 0.16 \times 7 = 1.12$
• For $x = 3$: $(3 - 1.60)^2 \times 3 = (1.40)^2 \times 3 = 1.96 \times 3 = 5.88$
• For $x = 4$: $(4 - 1.60)^2 \times 2 = (2.40)^2 \times 2 = 5.76 \times 2 = 11.52$

Step 2: Sum the squared deviations multiplied by frequency

Sum these values: $10.24 + 3.24 + 1.12 + 5.88 + 11.52 = 32.00$

Step 3: Calculate the sample variance

The formula for sample variance is $s^2 = \frac{\sum [(x - \bar{x})^2 \times f]}{n - 1}$, where $n = 25$ (total frequency).
Substitute: $s^2 = \frac{32.00}{25 - 1} = \frac{32.00}{24} \approx 1.3333$

Step 4: Calculate the sample standard deviation

Take the square root of the variance: $s = \sqrt{1.3333} \approx 1.15$

Answer:

s:
(a) $\bar{x} = \boxed{1.60}$
(b) $s = \boxed{1.15}$