Question

sc - 6 additional practice
in 1 - 5, use the data table.
cost of kick scooters at teds sports
$125 $135 $130 $140 $135 $154 $135 $130

1. what are the mean, median, and mode of these prices?

the mean is $129.5
the median is $135
the mode is $135

1. if ted sells the $154 scooter, how is the mean cost of scooters at the shop affected? explain.
2. if ted sells the $154 scooter, how are the median and mode cost of scooters at the shop affected? explain.
3. if ted sells the $154 scooter, how is the distribution of the cost of scooters at the shop affected? explain.
4. reasoning which would have a larger impact on the mean price of the remaining scooters: selling the $154 scooter or the $140 scooter? explain.

Explanation:

Step1: Calculate original mean

The original data set is $\{125, 135, 130, 140, 135, 154, 135, 130\}$. The sum of the data is $125 + 135+130 + 140+135+154+135+130=1084$. There are $n = 8$ data - points. The mean $\bar{x}_1=\frac{1084}{8}=135.5$.

Step2: Calculate new mean after selling $154$ scooter

After selling the $\$154$ scooter, the new data set is $\{125, 135, 130, 140, 135, 135, 130\}$. The sum of the new data is $125+135 + 130+140+135+135+130 = 930$. There are $n = 7$ data - points. The new mean $\bar{x}_2=\frac{930}{7}\approx132.86$. The mean decreases because the value of $154$ which is larger than the original mean is removed.

Step3: Check original median

Arrange the original data set in ascending order: $\{125, 130, 130, 135, 135, 135, 140, 154\}$. Since $n = 8$ (even), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values. The $\frac{8}{2}=4$th and $4 + 1=5$th values are $135$ and $135$, so the median $M_1=\frac{135 + 135}{2}=135$.

Step4: Check new median

Arrange the new data set in ascending order: $\{125, 130, 130, 135, 135, 135, 140\}$. Since $n = 7$ (odd), the median is the $(\frac{n + 1}{2})$th ordered value. The $(\frac{7+1}{2})=4$th value is $135$. The median remains the same.

Step5: Check original mode

The mode of the original data set is $135$ as it appears $3$ times, more frequently than any other value.

Step6: Check new mode

The mode of the new data set is still $135$ as it appears $3$ times, more frequently than any other value.

Step7: Analyze impact on distribution

The distribution was right - skewed (due to the outlier $154$). After removing the $154$ value, the distribution becomes more symmetric as the outlier is removed.

Step8: Analyze impact on mean for different sales

For selling the $\$154$ scooter:
Original sum $S_1 = 1084$, new sum $S_{2}=1084 - 154=930$, new mean $\bar{x}_2=\frac{930}{7}\approx132.86$.
For selling the $\$140$ scooter:
Original sum $S_1 = 1084$, new sum $S_{3}=1084-140 = 944$, new mean $\bar{x}_3=\frac{944}{7}\approx134.86$.
Selling the $\$154$ scooter has a larger impact on the mean because $154$ is further from the original mean ($135.5$) than $140$ is.

Answer:

1. Mean: $\$135.5$, Median: $\$135$, Mode: $\$135$
2. The mean decreases. The original mean is $\$135.5$ and the new mean is approximately $\$132.86$ because the value of $\$154$ (larger than the original mean) is removed.
3. The median remains the same at $\$135$ and the mode remains the same at $\$135$.
4. The distribution becomes more symmetric as the outlier ($154$) is removed.
5. Selling the $\$154$ scooter has a larger impact on the mean. Because $154$ is further from the original mean ($135.5$) than $140$ is.