Here is the step-by-step solution to your homework problem.
1) How many teens were surveyed?
To find the total number of teens, we need to add up the "Frequency" column. The frequency tells us how many teens fall into each category.
* Teens who downloaded 0-2 apps:
13
* Teens who downloaded 3-5 apps:
7
* Teens who downloaded 6-8 apps:
4
* Teens who downloaded 9-11 apps:
1
* Teens who downloaded 12-14 apps:
1
Calculation:
$$13 + 7 + 4 + 1 + 1 = 26$$
So,
26 teens were surveyed.
***
2) How many total apps were downloaded by this group of teens?
Since the data is grouped (for example, "0-2"), we don't know the exact number for each person. To estimate the total, we use the
midpoint of each group. The midpoint is the average of the lowest and highest number in that range.
Step A: Find the midpoint for each group
*
0-2: $(0 + 2) / 2 = \mathbf{1}$
*
3-5: $(3 + 5) / 2 = \mathbf{4}$
*
6-8: $(6 + 8) / 2 = \mathbf{7}$
*
9-11: $(9 + 11) / 2 = \mathbf{10}$
*
12-14: $(12 + 14) / 2 = \mathbf{13}$
Step B: Multiply the midpoint by the frequency (number of teens)
* $1 \text{ app} \times 13 \text{ teens} = \mathbf{13}$
* $4 \text{ apps} \times 7 \text{ teens} = \mathbf{28}$
* $7 \text{ apps} \times 4 \text{ teens} = \mathbf{28}$
* $10 \text{ apps} \times 1 \text{ teen} = \mathbf{10}$
* $13 \text{ apps} \times 1 \text{ teen} = \mathbf{13}$
Step C: Add the totals together
$$13 + 28 + 28 + 10 + 13 = 92$$
So, the estimated total is
92 apps.
***
3) Give one number that you think best represents the typical number of apps that a teen downloads in one month. Justify your answer.
A good number to represent the "typical" value is the
mean (average). We calculated the total estimated apps in question #2, and the total number of teens in question #1.
Calculation:
$$\text{Mean} = \frac{\text{Total Apps}}{\text{Total Teens}}$$
$$\text{Mean} = \frac{92}{26} \approx 3.54$$
Another option is the
mode, which is the most common group. The group "0-2" has the highest frequency (13 teens). The midpoint of that group is 1.
However, the mean (average) is usually the best single number to describe the whole group because it accounts for everyone, even the few teens who downloaded many apps.
Justification:
I choose
3.5 (or roughly 4). This is the average number of apps downloaded. It balances out the large group of teens who downloaded very few apps with the smaller group who downloaded many.
*(Note: If you prefer to use the Mode, you could say
1. Justification: This is the midpoint of the most frequent category, meaning more teens downloaded around 1 app than any other amount.)*
Final Answer:
1) 26 teens
2) 92 apps
3) 3.5 (This is the mean/average. It is calculated by dividing the total estimated apps (92) by the total number of teens (26). It represents the center of the data.)
Parent Tip: Review the logic above to help your child master the concept of frequency table and histogram worksheet.
