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	relative frequency	cumulative relative frequency
1	9
2	7
3	3
4	1

(a) construct a histogram of the data.
there are four histograms with the x - axis labeled number of movies (values 0, 1, 2, 3, 4) and y - axis labeled frequency (values 0, 2, 4, 6, 8). each histogram has bars for 0, 1, 2, 3, 4 movies with different heights.
(b) complete the columns of the chart. (enter exact numbers as integers, fractions, or decimals.)

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

Explanation:

Part (a)

To construct a histogram, we need to check the frequency of each number of movies:

• For 0 movies: frequency is 5
• For 1 movie: frequency is 9
• For 2 movies: frequency is 7
• For 3 movies: frequency is 3
• For 4 movies: frequency is 1

We look at the height of each bar in the histograms. The first histogram has:

• 0 movies: height 5 (matches frequency 5)
• 1 movie: height 9 (matches frequency 9)
• 2 movies: height 1 (but frequency is 7, so no)
• 3 movies: height 3 (matches frequency 3)
• 4 movies: height 7 (but frequency is 1, so no)

The second histogram:

• 0 movies: height 5 (matches)
• 1 movie: height 1 (frequency 9, no)
• 2 movies: height 7 (matches)
• 3 movies: height 3 (matches)
• 4 movies: height 9 (frequency 1, no)

The third histogram:

• 0 movies: height 5 (matches)
• 1 movie: height 9 (matches)
• 2 movies: height 7 (matches)
• 3 movies: height 3 (matches)
• 4 movies: height 1 (matches)

The fourth histogram:

• 0 movies: height 1 (frequency 5, no)
• 1 movie: height 3 (frequency 9, no)
• 2 movies: height 7 (matches)
• 3 movies: height 8 (frequency 3, no)
• 4 movies: height 5 (frequency 1, no)

So the correct histogram is the third one (the one with 0:5, 1:9, 2:7, 3:3, 4:1 heights).

Part (b)

Step 1: Calculate Relative Frequency

Relative Frequency is calculated as $\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Number of Students}}$. The total number of students is $5 + 9 + 7 + 3 + 1 = 25$.

• For 0 movies: $\frac{5}{25} = 0.2$
• For 1 movie: $\frac{9}{25} = 0.36$
• For 2 movies: $\frac{7}{25} = 0.28$
• For 3 movies: $\frac{3}{25} = 0.12$
• For 4 movies: $\frac{1}{25} = 0.04$

Step 2: Calculate Cumulative Relative Frequency

Cumulative Relative Frequency is the sum of relative frequencies up to that point.

• For 0 movies: cumulative relative frequency is the relative frequency of 0 movies, which is $0.2$ (since it's the first category, cumulative is the same as relative)
• For 1 movie: cumulative relative frequency is relative frequency of 0 + relative frequency of 1: $0.2 + 0.36 = 0.56$
• For 2 movies: cumulative relative frequency is $0.2 + 0.36 + 0.28 = 0.84$
• For 3 movies: cumulative relative frequency is $0.2 + 0.36 + 0.28 + 0.12 = 0.96$
• For 4 movies: cumulative relative frequency is $0.2 + 0.36 + 0.28 + 0.12 + 0.04 = 1.00$

Part (b) Table Completion

# of movies	Frequency	Relative Frequency	Cumulative Relative Frequency
1	9	$\frac{9}{25} = 0.36$	$0.2 + 0.36 = 0.56$
2	7	$\frac{7}{25} = 0.28$	$0.56 + 0.28 = 0.84$
3	3	$\frac{3}{25} = 0.12$	$0.84 + 0.12 = 0.96$
4	1	$\frac{1}{25} = 0.04$	$0.96 + 0.04 = 1.00$

Final Answers

(a)

The correct histogram is the third one (the one with 0:5, 1:9, 2:7, 3:3, 4:1 heights).

(b)

# of movies	Frequency	Relative Frequency	Cumulative Relative Frequency
1	9	$0.36$	$0.56$
2	7	$0.28$	$0.84$
3	3	$0.12$	$0.96$
4	1	$0.04$	$1.00$

Answer:

Part (a)

To construct a histogram, we need to check the frequency of each number of movies:

• For 0 movies: frequency is 5
• For 1 movie: frequency is 9
• For 2 movies: frequency is 7
• For 3 movies: frequency is 3
• For 4 movies: frequency is 1

We look at the height of each bar in the histograms. The first histogram has:

• 0 movies: height 5 (matches frequency 5)
• 1 movie: height 9 (matches frequency 9)
• 2 movies: height 1 (but frequency is 7, so no)
• 3 movies: height 3 (matches frequency 3)
• 4 movies: height 7 (but frequency is 1, so no)

The second histogram:

• 0 movies: height 5 (matches)
• 1 movie: height 1 (frequency 9, no)
• 2 movies: height 7 (matches)
• 3 movies: height 3 (matches)
• 4 movies: height 9 (frequency 1, no)

The third histogram:

• 0 movies: height 5 (matches)
• 1 movie: height 9 (matches)
• 2 movies: height 7 (matches)
• 3 movies: height 3 (matches)
• 4 movies: height 1 (matches)

The fourth histogram:

• 0 movies: height 1 (frequency 5, no)
• 1 movie: height 3 (frequency 9, no)
• 2 movies: height 7 (matches)
• 3 movies: height 8 (frequency 3, no)
• 4 movies: height 5 (frequency 1, no)

So the correct histogram is the third one (the one with 0:5, 1:9, 2:7, 3:3, 4:1 heights).

Part (b)

Step 1: Calculate Relative Frequency

Relative Frequency is calculated as $\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Number of Students}}$. The total number of students is $5 + 9 + 7 + 3 + 1 = 25$.

• For 0 movies: $\frac{5}{25} = 0.2$
• For 1 movie: $\frac{9}{25} = 0.36$
• For 2 movies: $\frac{7}{25} = 0.28$
• For 3 movies: $\frac{3}{25} = 0.12$
• For 4 movies: $\frac{1}{25} = 0.04$

Step 2: Calculate Cumulative Relative Frequency

Cumulative Relative Frequency is the sum of relative frequencies up to that point.

• For 0 movies: cumulative relative frequency is the relative frequency of 0 movies, which is $0.2$ (since it's the first category, cumulative is the same as relative)
• For 1 movie: cumulative relative frequency is relative frequency of 0 + relative frequency of 1: $0.2 + 0.36 = 0.56$
• For 2 movies: cumulative relative frequency is $0.2 + 0.36 + 0.28 = 0.84$
• For 3 movies: cumulative relative frequency is $0.2 + 0.36 + 0.28 + 0.12 = 0.96$
• For 4 movies: cumulative relative frequency is $0.2 + 0.36 + 0.28 + 0.12 + 0.04 = 1.00$

Part (b) Table Completion

# of movies	Frequency	Relative Frequency	Cumulative Relative Frequency
1	9	$\frac{9}{25} = 0.36$	$0.2 + 0.36 = 0.56$
2	7	$\frac{7}{25} = 0.28$	$0.56 + 0.28 = 0.84$
3	3	$\frac{3}{25} = 0.12$	$0.84 + 0.12 = 0.96$
4	1	$\frac{1}{25} = 0.04$	$0.96 + 0.04 = 1.00$

Final Answers

(a)

The correct histogram is the third one (the one with 0:5, 1:9, 2:7, 3:3, 4:1 heights).

(b)

# of movies	Frequency	Relative Frequency	Cumulative Relative Frequency
1	9	$0.36$	$0.56$
2	7	$0.28$	$0.84$
3	3	$0.12$	$0.96$
4	1	$0.04$	$1.00$