Question

in 6 - 8, use the line plot. ages of soloists in an orchestra

1. a new soloist is chosen from the orchestra. she is 33 years old. how are the mean, median, and mode affected by this addition? explain.
2. if the 33 - year - old soloist is added to the data set, how is the distribution affected? explain.
3. drop the two youngest soloists from the data set. why does this have a different effect on the distribution than adding a 33 - year - old soloist?

Explanation:

Step1: Analyze the mean

The mean is the sum of all values divided by the number of values. Since 33 is less than the current mean (as most values are clustered around 30 - 40), adding 33 will decrease the mean. Let the sum of the original $n$ values be $S$ and their mean be $\bar{x}=\frac{S}{n}$. After adding 33, the new mean $\bar{x}_{new}=\frac{S + 33}{n+1}$. Since 33 is relatively small compared to the original sum's contribution per - value, the new mean will be lower.

Step2: Analyze the median

Count the number of data - points in the original set. If $n$ is odd, the median is the middle - numbered value when the data is ordered. If $n$ is even, the median is the average of the two middle - numbered values. Since the number of data points is currently 20 (by counting the dots), $n = 20$ (even). The median is the average of the 10th and 11th ordered values. After adding a new value, $n=21$ (odd), and the new median will be the 11th ordered value. Since 33 is in the middle of the distribution, the new median will be 33, which is likely to be close to the original median (the average of the 10th and 11th values in the original 20 - value set), so the median will change slightly or remain approximately the same.

Step3: Analyze the mode

The mode is the most frequently occurring value. In the original data, the mode is around 36 - 38. Adding a single value of 33 does not change the most frequently occurring value(s), so the mode remains the same.

Step4: Analyze the distribution change in question 7

The original distribution is skewed slightly to the left (as there are some lower values like 12 and 14). Adding a 33 - year - old soloist makes the distribution more symmetric as 33 is more in the central part of the existing range of values. It fills in the gap between the lower and higher values, reducing the left - skew.

Step5: Analyze the distribution change in question 8

Dropping the two youngest soloists (12 and 14) removes the extreme lower values. This shifts the entire distribution to the right. It has a different effect than adding a 33 - year - old soloist because adding 33 fills a gap in the middle of the distribution, while dropping the two youngest values changes the lower - bound of the distribution, making all the remaining values relatively larger in the context of the new data set.

Answer:

1. Mean: Decreases. Median: Changes slightly or remains approximately the same. Mode: Remains the same.
2. The distribution becomes more symmetric.
3. Dropping the two youngest soloists shifts the distribution to the right. This is different from adding a 33 - year - old soloist because adding 33 fills a middle - range gap, while dropping the youngest soloists changes the lower - bound of the distribution.