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 data - value and frequency pairs

Assume we have data - values \(x_i\) and their corresponding frequencies \(f_i\) from the histogram. But since the frequencies are not given explicitly, we'll assume the height of each bar as the relative - frequency \(p_i\). Let's say for \(x = 0,1,\cdots,7\) the relative - frequencies are \(p_0,p_1,\cdots,p_7\).

Step2: Calculate the mean \(\mu\)

The formula for the mean of a discrete probability distribution is \(\mu=\sum_{i = 0}^{n}x_ip_i\).

Step3: Calculate the variance \(\sigma^{2}\)

The formula for the variance of a discrete probability distribution is \(\sigma^{2}=\sum_{i = 0}^{n}(x_i-\mu)^{2}p_i\).

Step4: Calculate the standard deviation \(\sigma\)

The standard deviation is the square - root of the variance, \(\sigma=\sqrt{\sigma^{2}}\).

Answer:

1. Shape of the histogram: The histogram is skewed right because the tail of the distribution extends to the right (the values on the right - hand side have lower frequencies and the peak is on the left side).
2. Mean \(\mu\): Without the actual frequency or relative - frequency values, we can't calculate the exact mean. But if we assume the relative - frequencies \(p_i\) for \(x_i\) from the histogram, we use \(\mu=\sum_{i = 0}^{7}x_ip_i\).
3. Variance \(\sigma^{2}\): Using the formula \(\sigma^{2}=\sum_{i = 0}^{7}(x_i - \mu)^{2}p_i\) after calculating the mean \(\mu\).
4. Standard deviation \(\sigma\): \(\sigma=\sqrt{\sigma^{2}}\), rounded to four decimal places as required.