Question

lifetime pets
which of the following is most likely true?
the mean and median are both in the interval 1 - 5.
the mean and median are both in the interval 6 - 10.
the mean is in the interval 6 - 10, and the median is in the interval 1 - 5.
the mean is in the interval 1 - 5, and the median is in the interval 6 - 10.

Explanation:

Step1: Analyze the distribution of data

The bar - graph shows that the interval \(1 - 5\) has the highest frequency. For the median:
Let the number of adults in the \(1 - 5\) interval be \(n_1\), in the \(6 - 10\) interval be \(n_2\), in the \(11 - 15\) interval be \(n_3\), and in the \(16 - 20\) interval be \(n_4\). Assume \(n_1>n_2>n_3>n_4\). The median is the middle value when the data is ordered. Since the \(1 - 5\) interval has the largest number of data points, the median is likely in the \(1 - 5\) interval.

Step2: Analyze the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{k}x_in_i}{\sum_{i=1}^{k}n_i}\), where \(x_i\) is the mid - point of the \(i\) - th interval. The mid - points: for the \(1 - 5\) interval \(x_1 = 3\), for the \(6 - 10\) interval \(x_2=8\), for the \(11 - 15\) interval \(x_3 = 13\), for the \(16 - 20\) interval \(x_4=18\).

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Since \(n_1\) is the largest, but \(n_2\) is also relatively large and the values of \(x_i\) for higher intervals (\(x_2,x_3,x_4\)) are larger than \(x_1\). If we assume \(n_1 = 14\), \(n_2=10\), \(n_3 = 5\), \(n_4=1\) (approximate values based on the bar - height)

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Answer:

The mean is in the interval \(6 - 10\), and the median is in the interval \(1 - 5\).