Question

a meteorologist is studying the monthly rainfall in a section of the brazilian rainforest. she recorded the monthly rainfall, in inches, for last year. they were:

10.4, 10.3, 11.7, 11.1, 8.0, 4.4, 2.6, 1.8, 2.5, 4.4, 7.3, 9.5

calculate the following for the data set:
mean:
median:
mode:
range:

which of the following is true if the data point 11.1 is corrected to 11.7?
the mean increases.
the median increases.
the data set has two modes.
the range increases.

Explanation:

Analyze the original data set

Using the Basic Statistics knowledge point
The original data set consists of 12 values:

$$ 10.4, 10.3, 11.7, 11.1, 8.0, 4.4, 2.6, 1.8, 2.5, 4.4, 7.3, 9.5 $$

Ordered from least to greatest:

$$ 1.8, 2.5, 2.6, 4.4, 4.4, 7.3, 8.0, 9.5, 10.3, 10.4, 11.1, 11.7 $$

The calculated statistics shown in the image are:

• Mean: \(7.0\)
• Median: \(7.65\) (average of the 6th and 7th values: \(\frac{7.3 + 8.0}{2} = 7.65\))
• Mode: \(4.4\)
• Range: \(11.7 - 1.8 = 9.9\)

Analyze the effect of correcting 11.1 to 11.7

Using the Basic Statistics knowledge point
The data point \(11.1\) is replaced by \(11.7\).
The new ordered data set is:

$$ 1.8, 2.5, 2.6, 4.4, 4.4, 7.3, 8.0, 9.5, 10.3, 10.4, 11.7, 11.7 $$

Let's evaluate each statement:

• The mean increases: Since one data value increases from \(11.1\) to \(11.7\), the sum of the data set increases. Therefore, the mean must increase. This statement is true.
• The median increases: The 6th and 7th values remain \(7.3\) and \(8.0\). The median remains \(\frac{7.3 + 8.0}{2} = 7.65\). This statement is false.
• The data set has two modes: The value \(4.4\) appears twice, and the value \(11.7\) now also appears twice. Thus, the new modes are \(4.4\) and \(11.7\). This statement is true.
• The range increases: The minimum remains \(1.8\) and the maximum remains \(11.7\). The range remains \(11.7 - 1.8 = 9.9\). This statement is false.

Determine the correct option

The question asks "Which of the following is true...". In standard single-choice questions of this format, we select the primary direct consequence on the measures of center. An increase in any data value always increases the mean. Let's list the options as presented in the image.

Answer:

• (A) The mean increases. (Correct answer)
• (B) The median increases.
• (C) The data set has two modes. (Correct answer)
• (D) The range increases.