Question

the numbers of courses taught per semester by a random sample of university professors are shown in the histogram. make a frequency distribution for the data. then use the table to estimate the sample mean and the sample standard deviation of the data set. make a frequency distribution for the data. the sample mean is \\( \bar { x } = 2.8 \\). (round to one decimal place as needed.) the sample standard deviation is \\( s = \square \\) (round to one decimal place as needed.)

Explanation:

Step1: Calculate the sum of \(f\)

$$ n=\sum f = 5 + 20+21 + 17=63 $$

Step2: Calculate \(\sum(f\cdot x)\)

$$ \sum(f\cdot x)=1\times5 + 2\times20+3\times21 + 4\times17=5+40 + 63+68=176 $$

Step3: Calculate \(\sum(f\cdot x^{2})\)

$$ \sum(f\cdot x^{2})=1^{2}\times5+2^{2}\times20 + 3^{2}\times21+4^{2}\times17=5 + 80+189+272=546 $$

Step4: Calculate the sample standard deviation formula \(s=\sqrt{\frac{n\sum(f\cdot x^{2})-(\sum(f\cdot x))^{2}}{n(n - 1)}}\)

Substitute \(n = 63\), \(\sum(f\cdot x)=176\), \(\sum(f\cdot x^{2})=546\) into the formula:

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Answer:

The sample standard deviation is \(s = 0.9\)