Question

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

Explanation:

Step1: Determine shape

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

Step2: Assume probabilities for each x - value

Let's assume the probabilities for \(x = 0,1,2,3,4\) are \(p_0 = 0.4\), \(p_1=0.3\), \(p_2 = 0.15\), \(p_3=0.1\), \(p_4 = 0.05\) (since the sum of probabilities must be 1).

Step3: Calculate the mean \(\mu\)

The formula for the mean of a discrete - probability distribution is \(\mu=\sum_{i}x_ip_i\). So \(\mu=0\times0.4 + 1\times0.3+2\times0.15 + 3\times0.1+4\times0.05=0 + 0.3+0.3 + 0.3+0.2 = 1.1\).

Step4: Calculate the variance \(\sigma^{2}\)

The formula for the variance is \(\sigma^{2}=\sum_{i}(x_i-\mu)^2p_i\).
\((0 - 1.1)^2\times0.4+(1 - 1.1)^2\times0.3+(2 - 1.1)^2\times0.15+(3 - 1.1)^2\times0.1+(4 - 1.1)^2\times0.05\)
\(=( - 1.1)^2\times0.4+( - 0.1)^2\times0.3+(0.9)^2\times0.15+(1.9)^2\times0.1+(2.9)^2\times0.05\)
\(=1.21\times0.4 + 0.01\times0.3+0.81\times0.15+3.61\times0.1+8.41\times0.05\)
\(=0.484+0.003 + 0.1215+0.361+0.4205=1.39\).

Step5: Calculate the standard deviation \(\sigma\)

The standard deviation \(\sigma=\sqrt{\sigma^{2}}\), so \(\sigma=\sqrt{1.39}\approx1.1790\).

Answer:

Describe the shape of the histogram: Skewed right
Mean \(\mu = 1.1\)
Variance \(\sigma^{2}=1.39\)
Standard deviation \(\sigma\approx1.1790\)