Question

unit 1 - exploring one variable data
hw 4 - socs and comparing
name: yana j
karley works as a dominoes delivery driver and on friday night, she recorded how many tips she got from each of her deliveries. the histogram below shows the distribution of the 33 deliveries.
a) describe the distribution of tips.
b) one of the tip amounts was $9. if this tip had been $14 instead of $9, describe what effect this would have on the following measures of center. justify your answer.
mean:
median:
c) describe how you would estimate the median of this distribution.

Explanation:

Step1: Analyze distribution shape

The distribution is skewed right. There is a peak around the 6 - 8 dollar tip amount range. The left - hand side of the distribution has fewer values and the tail extends to the right with some high - value outliers (e.g., tips of 22 dollars).

Step2: Analyze effect on mean

The mean is the sum of all values divided by the number of values. If a value increases (from 9 to 14), the sum of all values increases. Since the number of values ($n = 33$) remains the same, the mean will increase. Mathematically, let $S$ be the sum of all tip amounts before the change. The original mean $\bar{x}_1=\frac{S}{33}$. After changing a 9 to 14, the new sum is $S - 9+14=S + 5$, and the new mean $\bar{x}_2=\frac{S + 5}{33}=\bar{x}_1+\frac{5}{33}$.

Step3: Analyze effect on median

There are $n = 33$ data points. The median is the 17th ordered value. Changing a single value from 9 to 14 does not change the position of the 17th value in the ordered list of tip amounts. So the median will remain the same.

Step4: Describe median estimation

First, order the data (in terms of frequency in the histogram). Since $n = 33$, the median is the 17th value. Start from the left - most bar and accumulate frequencies. The first bar (0 - 2 dollars) has a frequency of 1, the second (2 - 4 dollars) has a frequency of 3, the third (4 - 6 dollars) has a frequency of 8. The sum of these frequencies is $1+3 + 8=12$. The next bar (6 - 8 dollars) has a frequency of 14. Since $12<17$ and $12 + 14>17$, the median lies within the 6 - 8 dollar interval.

Answer:

a) The distribution of tips is skewed right with a peak around 6 - 8 dollars and some high - value outliers.
b) Mean: The mean will increase because increasing a data value increases the sum of all data values while the number of data values remains the same. Median: The median will remain the same because changing a single value does not change the position of the middle (17th) value in the ordered data set.
c) First, identify that the median is the 17th value for $n = 33$ data points. Then, accumulate frequencies starting from the left - most bar of the histogram. The median lies within the bar where the cumulative frequency first exceeds 17, which is the 6 - 8 dollar interval.