Question

1. select all the distribution shapes for which it is most often appropriate to use the mean. a. bell - shaped b. bimodal c. skewed d. symmetric e. uniform 2. for which distribution shape is it usually appropriate to use the median when summarizing the data? a. bell - shaped b. skewed c. symmetric d. uniform 3. the number of writing instruments in some teachers desks is displayed in the dot plot. which is greater, the mean or the median? explain your reasoning using the shape of the distribution. 4. a student has these scores on their assignments. the teacher is considering dropping a lowest score. what effect does eliminating the lowest value, 0, from the data set have on the mean and median? (lesson 1 - 9) 0, 40, 60, 70, 75, 80, 85, 95, 95, 100

Explanation:

Step1: Recall when mean is appropriate

The mean is most appropriate for symmetric - shaped distributions such as bell - shaped and uniform distributions. Bell - shaped (normal) distributions are symmetric and the mean represents the center well. Uniform distributions are also symmetric and the mean is a good measure of center. Symmetric distributions in general are suitable for the mean. So for question 1, the appropriate shapes are bell - shaped, symmetric, and uniform.

Step2: Recall when median is appropriate

The median is a better measure for skewed distributions. In skewed distributions, the mean is pulled in the direction of the skew, while the median represents the middle value and is not affected by extreme values as much. So for question 2, the answer is skewed.

Step3: Analyze dot - plot for question 3

The dot - plot is symmetric. In a symmetric distribution, the mean and median are approximately equal. Since the distribution is symmetric around the values 9 and 10, the mean and median are close, but without exact calculation, we can't say which is greater precisely. However, if we assume a perfect symmetric distribution, they are equal.

Step4: Analyze effect on mean and median for question 4

Mean before:

The mean of the data set \(0,40,60,70,75,80,85,95,95,100\) is \(\bar{x}_1=\frac{0 + 40+60+70+75+80+85+95+95+100}{10}=\frac{600}{10} = 60\).

Mean after:

After removing 0, the new mean is \(\bar{x}_2=\frac{40+60+70+75+80+85+95+95+100}{9}=\frac{600}{9}\approx66.67\). So the mean increases.

Median before:

The data set has 10 values. The median is the average of the 5th and 6th ordered values. The 5th value is 75 and the 6th value is 80, so the median \(M_1=\frac{75 + 80}{2}=77.5\).

Median after:

The new data set has 9 values. The median is the 5th ordered value, which is 80. So the median increases slightly.

Answer:

1. A. bell - shaped, D. symmetric, E. uniform
2. B. skewed
3. They are approximately equal (due to symmetric distribution)
4. Mean increases, Median increases