Question

listed below are the amounts (dollars) it costs for marriage proposal packages at different baseball stadiums. are there any outliers, and are they likely to have much of an effect on the measures of variation? find the range, variance, and standard deviation for the given sample data. include appropriate units in the results.

38 45 50 60 60 75 90 160 195 200 265 375 400 1750 2750

the range of the sample data is 2712 dollars. (type an integer or a decimal. do not round.)

the standard deviation of the sample data is 770 dollars. (round to one decimal place as needed.)

the variance of the sample data is 592751.9 dollars². (round to one decimal place as needed.)

are there any outliers and, if so, are they likely to have much of an effect on the measures of variation?

a. no, there are not any outliers.

b. yes, the largest amounts are much higher than the rest of the data, and appear to be outliers. it is likely that these are having a large effect on the measures of variation.

c. yes, the largest amounts are much higher than the rest of the data, and appear to be outliers. it is not likely that these are having a large effect on the measures of variation.

d. yes, the smallest amounts are much lower than the rest of the data, and appear to be outliers. it is not likely that these are having a large effect on the measures of variation.

Explanation:

Outlier Analysis

To determine outliers, we use the interquartile range (IQR) method. First, order the data: \(38, 45, 50, 60, 60, 75, 90, 160, 195, 200, 265, 375, 400, 1750, 2750\).

1. Find Quartiles:

• Median (Q2) of 15 data points is the 8th value: \(160\).
• Lower half: \(38, 45, 50, 60, 60, 75, 90\) (median Q1 = 60).
• Upper half: \(195, 200, 265, 375, 400, 1750, 2750\) (median Q3 = 375).

1. Calculate IQR:

\(IQR = Q3 - Q1 = 375 - 60 = 315\).

1. Outlier Bounds:

• Lower bound: \(Q1 - 1.5 \times IQR = 60 - 1.5(315) = -412.5\) (no lower outliers, as all data > -412.5).
• Upper bound: \(Q3 + 1.5 \times IQR = 375 + 1.5(315) = 847.5\).

Data points \(1750\) and \(2750\) exceed \(847.5\), so they are outliers. These large values (much higher than most data) will inflate measures of variation (range, variance, standard deviation), as seen in the given range (\(2712\)), variance (\(592751.9\)), and standard deviation (\(770\))—all heavily influenced by the outliers.

Answer:

B. Yes, the largest amounts are much higher than the rest of the data, and appear to be outliers. It is likely that these are having a large effect on the measures of variation.