cara computes the mean and variance for the set 87, 46, 90, 78, and 89. she finds the mean to be 78. her…

cara computes the mean and variance for the set 87, 46, 90, 78, and 89. she finds the mean to be 78. her steps for finding the variance are shown below.\n$sigma^{2}=\frac{(87 - 78)^{2}+(46 - 78)^{2}+(90 - 78)^{2}+(78 - 78)^{2}+(89 - 78)^{2}}{5}$\n$sigma^{2}=\frac{(9)^{2}-(32)^{2}+(12)^{2}+0^{2}+(11)^{2}}{5}$\n$sigma^{2}=\frac{81 - 1024+144 + 0+121}{5}$\n$sigma^{2}=\frac{-678}{5}=-135.6$\nwhat is the first error cara made in computing the variance?\nshe did not find the correct difference of 89 - 78.\nshe divided by 5 instead of 4.\nshe put the negative sign for the 32 outside the parentheses.\nshe forgot to take the square root of - 135.6.

cara computes the mean and variance for the set 87, 46, 90, 78, and 89. she finds the mean to be 78. her steps for finding the variance are shown below.\n$sigma^{2}=\frac{(87 - 78)^{2}+(46 - 78)^{2}+(90 - 78)^{2}+(78 - 78)^{2}+(89 - 78)^{2}}{5}$\n$sigma^{2}=\frac{(9)^{2}-(32)^{2}+(12)^{2}+0^{2}+(11)^{2}}{5}$\n$sigma^{2}=\frac{81 - 1024+144 + 0+121}{5}$\n$sigma^{2}=\frac{-678}{5}=-135.6$\nwhat is the first error cara made in computing the variance?\nshe did not find the correct difference of 89 - 78.\nshe divided by 5 instead of 4.\nshe put the negative sign for the 32 outside the parentheses.\nshe forgot to take the square root of - 135.6.

Answer

Answer: She put the negative sign for the 32 outside the parentheses.

Explanation:

Step1: Recall variance formula

The formula for population variance $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}$, where $x_{i}$ are data - points, $\mu$ is the mean and $n$ is the number of data - points.

Step2: Analyze Cara's work

When calculating $(46 - 78)^{2}$, the correct result is $(-32)^{2}=1024$. But Cara wrote $(9)^{2}-(32)^{2}+(12)^{2}+0^{2}+(11)^{2}$ in the second step. She wrongly put the negative sign of 32 outside the parentheses. It should be $(9)^{2}+(- 32)^{2}+(12)^{2}+0^{2}+(11)^{2}$ since squaring a negative number gives a positive result. So the first error Cara made is that she put the negative sign for the 32 outside the parentheses.