Question

find the (a) mean, (b) median, (c) mode, and (d) midrange for the data and then (e) answer the given question. listed below are the amounts (dollars) it costs for marriage proposal packages at different sports venues. are there any outliers? 39 70 70 70 75 75 85 95 100 125 175 175 250 259 275 275 300 375 425 450 450 450 450 2000 3000 a. the mode(s) is(are) $450 (type an integer or a decimal. do not round. use a comma to separate answers as needed.) b. there is no mode d. find the midrange. the midrange is $1519.50 (type an integer or a decimal rounded to two decimal places as needed.) e. are there any outliers? choose the correct answer below a. the values $39, $2000, and $3000 appear to be outliers b. the values $2000 and $3000 appear to be outliers c. there do not appear to be any outliers d. the values $39 and $3000 appear to be outliers

Explanation:

Step1: Recall mid - range formula

The mid - range is calculated as $\frac{\text{Minimum value}+\text{Maximum value}}{2}$.

Step2: Identify minimum and maximum values

The minimum value in the data set is $39$ and the maximum value is $3000$.

Step3: Calculate mid - range

Mid - range=$\frac{39 + 3000}{2}=\frac{3039}{2}=1519.50$.

For outliers, we can use the inter - quartile range (IQR) method. First, we need to find the first quartile ($Q_1$), the third quartile ($Q_3$), and the IQR.

1. Arrange data in ascending order:

$39,70,70,70,75,75,85,95,100,125,175,175,250,259,275,275,300,375,425,450,450,450,450,2000,3000$

1. Find the median ($Q_2$):

There are $n = 25$ data points. The median is the $(\frac{n + 1}{2})$ - th value. So, the median is the $13$ - th value, which is $250$.

1. Find $Q_1$:

The lower half of the data has $12$ values. The median of the lower half is the average of the $6$ - th and $7$ - th values. So, $Q_1=\frac{75 + 85}{2}=80$.

1. Find $Q_3$:

The upper half of the data has $12$ values. The median of the upper half is the average of the $19$ - th and $20$ - th values. So, $Q_3=\frac{425+450}{2}=437.5$.

1. Calculate IQR:

$IQR = Q_3−Q_1=437.5 - 80 = 357.5$.

1. Define outlier boundaries:

Lower boundary=$Q_1-1.5\times IQR=80-1.5\times357.5=80 - 536.25=-456.25$
Upper boundary=$Q_3 + 1.5\times IQR=437.5+1.5\times357.5=437.5+536.25 = 973.75$

Values less than the lower boundary or greater than the upper boundary are outliers. The values $2000$ and $3000$ are greater than $973.75$, so they are outliers.

Answer:

a. Mean:
Sum of all values: $39+70+70+70+75+75+85+95+100+125+175+175+250+259+275+275+300+375+425+450+450+450+450+2000+3000 = 9169$
Mean=$\frac{9169}{25}=366.76$
b. Median: $250$
c. Mode: $450$
d. Mid - range: $1519.50$
e. B. The values $\$2000$ and $\$3000$ appear to be outliers