finding variance\ncara computes the mean and variance for the set 87, 46, 90, 78, and 89. she finds the mean…

finding variance\ncara 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?

finding variance\ncara 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?

Answer

Explanation:

Step1: Analyze the variance - formula step

The formula for the variance of a sample $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$, where $x_{i}$ are the data - points and $\bar{x}$ is the mean and $n$ is the number of data - points. When calculating $(46 - 78)^{2}$, Cara made an error. $(46-78)=-32$, so $(46 - 78)^{2}=(-32)^{2}=1024$, but Cara wrote it as $-(32)^{2}=- 1024$. The correct calculation of the variance should be: [ \begin{align*} \sigma^{2}&=\frac{(87 - 78)^{2}+(46 - 78)^{2}+(90 - 78)^{2}+(78 - 78)^{2}+(89 - 78)^{2}}{5}\ &=\frac{9^{2}+(-32)^{2}+12^{2}+0^{2}+11^{2}}{5}\ &=\frac{81 + 1024+144 + 0+121}{5}\ &=\frac{1370}{5}\ & = 274 \end{align*} ]

Answer:

Cara incorrectly calculated $(46 - 78)^{2}$ as $-(32)^{2}$ instead of $(-32)^{2}$.