listed below are the playing times (in seconds) of songs that were popular at the time of this writing. find the (a) mean, (b) median, (c) mode, and (d) midrange for the given sample data. is there one time that is very different from the others?
452 237 226 249 249 296 280 227 239 209 263 235 209 261 260 257
a. the mean is 259.3 seconds. (round to one decimal place as needed.)
b. the median is 249 seconds. (round to one decimal place as needed.)
c. select the correct choice below and fill in any answer boxes in your choice.
a. the mode is 209, 249 seconds. (use a comma to separate answers as needed. round to one decimal place as needed.)
b. there is no mode.
d. the midrange is 330.5 seconds. (round to one decimal place as needed.)
is there one time that is very different from the others?
a. no; all the times are not very different from each other.
b. yes; the time of 209 seconds is very different from the others.
c. yes; the time of 296 seconds is very different from the others.
d. yes; the time of 452 seconds is very different from the others.

Question

Accepted Answer

### Part a: Mean Calculation
# Explanation:
## Step1: Sum all the data points
First, we list out all the playing times (in seconds): 452, 237, 226, 249, 249, 296, 280, 227, 239, 209, 263, 235, 209, 261, 260, 257.
Now, we sum them up:
\[
\begin{align*}
&452 + 237 + 226 + 249 + 249 + 296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 452+(237 + 226)+(249 + 249)+296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 452 + 463+498 + 296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(452 + 463)+498 + 296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 915+498 + 296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(915 + 498)+296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 1413+296 + 280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(1413 + 296)+280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 1709+280 + 227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(1709 + 280)+227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 1989+227 + 239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(1989 + 227)+239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 2216+239 + 209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(2216 + 239)+209 + 263 + 235 + 209 + 261 + 260 + 257\
=& 2455+209 + 263 + 235 + 209 + 261 + 260 + 257\
=&(2455 + 209)+263 + 235 + 209 + 261 + 260 + 257\
=& 2664+263 + 235 + 209 + 261 + 260 + 257\
=&(2664 + 263)+235 + 209 + 261 + 260 + 257\
=& 2927+235 + 209 + 261 + 260 + 257\
=&(2927 + 235)+209 + 261 + 260 + 257\
=& 3162+209 + 261 + 260 + 257\
=&(3162 + 209)+261 + 260 + 257\
=& 3371+261 + 260 + 257\
=&(3371 + 261)+260 + 257\
=& 3632+260 + 257\
=&(3632 + 260)+257\
=& 3892+257\
=& 4149
\end{align*}
\]
## Step2: Count the number of data points
We count the number of playing times. Let's list them again and count: 452, 237, 226, 249, 249, 296, 280, 227, 239, 209, 263, 235, 209, 261, 260, 257. So there are $ n = 16 $ data points.
## Step3: Calculate the mean
The formula for the mean $ \bar{x} $ is $ \bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n} $, where $ \sum_{i=1}^{n}x_{i} $ is the sum of the data points and $ n $ is the number of data points.
We have $ \sum_{i = 1}^{n}x_{i}=4149 $ and $ n = 16 $. So,
$$
\bar{x}=\frac{4149}{16}=259.3125\approx259.3
$$
# Answer: 259.3 seconds

### Part b: Median Calculation
# Explanation:
## Step1: Arrange the data in ascending order
First, we sort the data points from smallest to largest:
209, 209, 226, 227, 235, 237, 239, 249, 249, 257, 260, 261, 263, 280, 296, 452
## Step2: Determine the position of the median
Since the number of data points $ n = 16 $ (an even number), the median is the average of the $ \frac{n}{2} $-th and $ (\frac{n}{2}+1) $-th values.
$ \frac{n}{2}=\frac{16}{2}=8 $ and $ \frac{n}{2}+1 = 9 $
## Step3: Find the 8th and 9th values and calculate the median
From the sorted data:
1st: 209, 2nd: 209, 3rd: 226, 4th: 227, 5th: 235, 6th: 237, 7th: 239, 8th: 249, 9th: 249, 10th: 257, 11th: 260, 12th: 261, 13th: 263, 14th: 280, 15th: 296, 16th: 452
The 8th value is 249 and the 9th value is 249. The median $ M $ is $ \frac{249 + 249}{2}=\frac{498}{2}=249 $
# Answer: 249 seconds

### Part c: Mode Calculation
# Explanation:
The mode is the value(s) that appear most frequently in the data set.
We count the frequency of each data point:
- 209: appears 2 times
- 226: appears 1 time
- 227: appears 1 time
- 235: appears 1 time
- 237: appears 1 time
- 239: appears 1 time
- 249: appears 2 times
- 257: appears 1 time
- 260: appears 1 time
- 261: appears 1 time
- 263: appears 1 time
- 280: appears 1 time
- 296: appears 1 time
- 452: appears 1 time

The values 209 and 249 both appear 2 times, which is more frequent than any other value.
# Answer: 209, 249 seconds

### Part d: Midrange Calculation
# Explanation:
## Step1: Identify the minimum and maximum values
From the data set, the minimum value $ \text{min} = 209 $ and the maximum value $ \text{max}=452 $
## Step2: Calculate the midrange
The formula for the midrange is $ \text{Midrange}=\frac{\text{min}+\text{max}}{2} $
So, $ \text{Midrange}=\frac{209 + 452}{2}=\frac{661}{2}=330.5 $
# Answer: 330.5 seconds

### Part e: Outlier Check
# Explanation:
To check if there is a time very different from the others, we look at the data points. Most of the data points are in the range of 200 - 300 seconds, except 452 seconds. 452 is much larger than the other values (the next largest is 296, and 452 - 296 = 156, which is a significant difference). The other values like 209, 249, 296 are within a reasonable range relative to each other. So the time of 452 seconds is very different from the others.
# Answer: D. Yes; the time of 452 seconds is very different from the others.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part a: Mean Calculation

Step1: Sum all the data points

Step2: Count the number of data points

Step3: Calculate the mean

Step1: Arrange the data in ascending order

Step2: Determine the position of the median

Step3: Find the 8th and 9th values and calculate the median

Answer:

Part b: Median Calculation