Let's work through the questions based on the image you provided. I'll solve each part step by step.

Data Provided:

• Number of Movies: 1, 2, 3, 4, 5, 6
• Frequency: 7, 2, 7, 5, 3, 1

Step 1: Expand the Data

Let's list out all the data points:

• 1 movie: 7 respondents
• 2 movies: 2 respondents
• 3 movies: 7 respondents
• 4 movies: 5 respondents
• 5 movies: 3 respondents
• 6 movies: 1 respondent

So, the dataset is: $1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6$

(a) Mean

The mean is calculated as: $\text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}}$

Sum of all values: $(7 \times 1) + (2 \times 2) + (7 \times 3) + (5 \times 4) + (3 \times 5) + (1 \times 6) = 7 + 4 + 21 + 20 + 15 + 6 = 73$

Total number of values: $7 + 2 + 7 + 5 + 3 + 1 = 25$

Thus: $\text{Mean} = \frac{73}{25} = 2.92$

(b) Median

To find the median, sort the data (already sorted) and find the middle value. Since there are 25 data points, the median is the 13th value in the list.

The 13th value is 3.

(c) Sample Standard Deviation

The formula for sample standard deviation $s$ is: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ where $\bar{x}$ is the mean, and $n$ is the number of observations.

Calculating each squared difference:

• For $x = 1$, $(1 - 2.92)^2 = 3.6864$, and there are 7 such terms.
• For $x = 2$, $(2 - 2.92)^2 = 0.8464$, and there are 2 such terms.
• For $x = 3$, $(3 - 2.92)^2 = 0.0064$, and there are 7 such terms.
• For $x = 4$, $(4 - 2.92)^2 = 1.1664$, and there are 5 such terms.
• For $x = 5$, $(5 - 2.92)^2 = 4.3364$, and there are 3 such terms.
• For $x = 6$, $(6 - 2.92)^2 = 9.5064$, and there is 1 such term.

Summing these gives: $7 \times 3.6864 + 2 \times 0.8464 + 7 \times 0.0064 + 5 \times 1.1664 + 3 \times 4.3364 + 1 \times 9.5064 = 25.8048 + 1.6928 + 0.0448 + 5.832 + 13.0092 + 9.5064 = 55.89$

$s = \sqrt{\frac{55.89}{24}} = \sqrt{2.32875} \approx 1.5259$

(d) What percent of the respondents watched at least 3 movies?

The number of respondents who watched at least 3 movies is: $7 (\text{watched 3 movies}) + 5 (\text{watched 4 movies}) + 3 (\text{watched 5 movies}) + 1 (\text{watched 6 movies}) = 16$ Percentage: $\frac{16}{25} \times 100 = 64\%$

(e) 40% of all respondents watched fewer than how many movies?

First, calculate 40% of the total respondents: $0.4 \times 25 = 10$

Now, count respondents starting from the fewest movies until the sum reaches 10:

• 7 watched 1 movie
• 2 watched 2 movies

So, 9 respondents watched 2 or fewer movies. Therefore, fewer than 3 movies.

Final Answers:

a. Mean: 2.9200
b. Median: 3
c. Sample Standard Deviation: 1.5259
d. Percent watching at least 3 movies: 64%
e. 40% watched fewer than: 3 movies

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the mode of the dataset?
2. How would the sample standard deviation change if the highest frequency were at 6 movies?
3. How is the variance related to the standard deviation?
4. Can you find the range of the dataset?
5. What would happen to the mean if all frequencies doubled?

Tip: Always verify if the dataset is sorted when finding the median, especially if provided frequencies or classes are non-sequential.