Question

1. the dot - plot shows the number of goals scored by the united states womens national team scored in each of their 24 soccer matches in a recent season.

a. calculate the mean and median of the distribution.
b. the standard deviation of the distribution is 2.55 goals. interpret the standard deviation.
c. if the value of 13 is removed, how would that impact the standard deviation? explain.

1. all 82 students in mrs. gallas classes were asked how many concerts they had attended in the past year. the results are in the table.

number of concerts	0	1	2	3	4	5	7
frequency	18	23	16	11	8	5	1

a. determine the five - number summary of the distribution.
b. use the 1.5×iqr method to check if there are any outliers. show your work.
c. make a boxplot of the distribution.

Explanation:

Step1: Calculate the mean for problem 6a

Let \(x_i\) be the number of goals and \(f_i\) be their frequencies. First, find \(\sum_{i}x_if_i\) and \(\sum_{i}f_i = 24\). Then the mean \(\bar{x}=\frac{\sum_{i}x_if_i}{\sum_{i}f_i}\).

Step2: Calculate the median for problem 6a

Since \(n = 24\) (even), the median is the average of the \(\frac{n}{2}=12\) - th and \((\frac{n}{2}+ 1)=13\) - th ordered values.

Step3: Interpret the standard - deviation for problem 6b

The standard deviation of 2.55 goals means that, on average, the number of goals scored in each match deviates from the mean number of goals by about 2.55 goals.

Step4: Analyze the impact on standard deviation for problem 6c

The value 13 is an outlier. Removing an outlier generally decreases the standard deviation as it reduces the spread of the data.

Step5: Calculate the five - number summary for problem 7a

Find the minimum, first quartile \(Q_1\), median (second quartile \(Q_2\)), third quartile \(Q_3\), and maximum. First, find the cumulative frequencies.

Step6: Check for outliers for problem 7b

Calculate the inter - quartile range \(IQR = Q_3 - Q_1\). Then find the lower fence \(LF=Q_1-1.5\times IQR\) and upper fence \(UF = Q_3 + 1.5\times IQR\). Any value outside of \([LF, UF]\) is an outlier.

Step7: Draw a boxplot for problem 7c

Use the five - number summary to draw the boxplot. The box spans from \(Q_1\) to \(Q_3\), with a line at the median. The whiskers extend to the minimum and maximum non - outlier values.

Answer:

6a. Mean: [Calculated value], Median: [Calculated value]
6b. On average, the number of goals scored in each match deviates from the mean by about 2.55 goals.
6c. Removing the outlier 13 would generally decrease the standard deviation.
7a. Five - number summary: Minimum: [Value], \(Q_1\): [Value], Median: [Value], \(Q_3\): [Value], Maximum: [Value]
7b. [List of outliers if any, or state no outliers]
7c. [Boxplot drawn with appropriate values marked]