Question

describe the shape of the histogram. skewed left skewed right roughly symmetric find the mean. $mu = 4.23$ find the variance. $sigma^{2}=2.03612$ find the standard deviation. round to four decimal places, if necessary. $sigma = 1.4269$

Explanation:

Step1: Identify shape

The tail of the histogram extends to the left, so it is skewed left.

Step2: Calculate mean formula

The mean $\mu=\sum_{i}x_ip_i$. Assume the mid - points of the intervals are $x_i$ and the heights of the bars (probabilities) are $p_i$. Let's say the mid - points are $x_1 = 0.5,x_2 = 1.5,x_3 = 2.5,x_4 = 3.5,x_5 = 4.5,x_6 = 5.5,x_7 = 6.5$ and the corresponding probabilities $p_1,p_2,p_3,p_4,p_5,p_6,p_7$ read from the bar heights. $\mu=(0.5\times p_1)+(1.5\times p_2)+(2.5\times p_3)+(3.5\times p_4)+(4.5\times p_5)+(5.5\times p_6)+(6.5\times p_7)$.

Step3: Calculate variance formula

The variance $\sigma^{2}=\sum_{i}(x_i - \mu)^2p_i$. First find the difference between each $x_i$ and the mean $\mu$, square it, and then multiply by the corresponding probability $p_i$ and sum them up.

Step4: Calculate standard deviation formula

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. Take the square - root of the variance.

Answer:

Describe the shape of the histogram: Skewed left
Find the mean: Calculate using $\mu=\sum_{i}x_ip_i$
Find the variance: Calculate using $\sigma^{2}=\sum_{i}(x_i - \mu)^2p_i$
Find the standard deviation: Calculate using $\sigma=\sqrt{\sigma^{2}}$ (round to four decimal places if necessary)