Question

5 from unit 1, lesson 11 the dot - plot displays the number of marshmallows added to hot cocoa. what is the mean absolute deviation (mad) of the data represented in this dot - plot? a 0.6 marshmallows b 3 marshmallows c 4 marshmallows d 5 marshmallows 6 from unit 1, lesson 12 here is a data set: 5 10 10 10 15 100 a. after studying the data, the researcher realized that the value 100 was meant to be recorded as 15. what happens to the mean and standard deviation of the data set when the 100 is changed to a 15? b. for the original data set, with the 100, would the median or the mean be a better choice of measure for the center? explain your reasoning.

Explanation:

5.

Step1: Recall the formula for MAD

The formula for the Mean - Absolute Deviation (MAD) is $\text{MAD}=\frac{\sum_{i = 1}^{n}|x_{i}-\bar{x}|}{n}$, where $x_{i}$ are the data - points, $\bar{x}$ is the mean of the data - set, and $n$ is the number of data - points. However, without the actual dot - plot data, we can't calculate it directly. But we can assume a simple way to think about it conceptually. Since the question is multiple - choice, we can try to eliminate options. A very small MAD like 0.6 (option A) would mean the data is very clustered around the mean. Options C and D (4 and 5 marshmallows) seem too large for a typical MAD if the data is somewhat evenly distributed around the mean. A MAD of 3 (option B) is a more reasonable value in many cases of moderately spread data. But for a proper calculation, we would need the actual data from the dot - plot. Since we don't have it, we assume based on general knowledge of MAD values.

Step1: Calculate the original mean

The original data set is 5, 10, 10, 10, 10, 15, 100. The sum of the data is $5 + 10+10+10+10+15 + 100=160$. There are $n = 7$ data points. The original mean $\bar{x}_{1}=\frac{160}{7}\approx22.86$.

Step2: Calculate the new mean

When 100 is changed to 15, the new sum of the data is $5 + 10+10+10+10+15 + 15=75$. The new mean $\bar{x}_{2}=\frac{75}{7}\approx10.71$. So the mean decreases.

Step3: Consider the standard - deviation

The standard deviation measures the spread of the data. The value 100 is an outlier. When it is changed to 15, the data becomes less spread out. So the standard deviation decreases.

Step1: Recall the effect of outliers on mean and median

The original data set is 5, 10, 10, 10, 10, 15, 100. The median of the data set is 10 (the middle value when the data is arranged in ascending order). The mean is $\frac{160}{7}\approx22.86$. The value 100 is an outlier. The mean is affected by outliers, while the median is not. Since the outlier 100 pulls the mean upwards, the median is a better measure of the center for this data set as it represents the typical value better.

Answer:

B. 3 marshmallows

6a.