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.
\begin{array}{c|c}
x & f \\
\hline
1 & 4 \\
2 & 20 \\
3 & 21 \\
4 & 15
\end{array}

the sample mean is \bar{x} = 2.8.
(round to one decimal place as needed.)

the sample standard deviation is s = .
(round to one decimal place as needed.)

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.
\begin{array}{c|c}
x & f \\
\hline
1 & 4 \\
2 & 20 \\
3 & 21 \\
4 & 15
\end{array}

the sample mean is \bar{x} = 2.8.
(round to one decimal place as needed.)

the sample standard deviation is s =  .
(round to one decimal place as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Frequency Distribution", "Mean Calculation", "Sample Standard Deviation" ], "new_concepts": [], "current_concepts": [ "Frequency Distribution", "Mean Calculation", "Sample Standard Deviation" ] } ## Identify the given frequency distribution Using the *Frequency Distribution* knowledge point $$ \begin{array}{c|c} x & f \ \hline 1 & 4 \ 2 & 20 \ 3 & 21 \ 4 & 15 \ \end{array} $$ ## Calculate the sample size and sum Using the *Mean Calculation* knowledge point $$ \begin{aligned} n &= \sum f = 4 + 20 + 21 + 15 = 60 \ \sum x \cdot f &= (1 \cdot 4) + (2 \cdot 20) + (3 \cdot 21) + (4 \cdot 15) \ &= 4 + 40 + 63 + 60 = 167 \ \bar{x} &= \frac{167}{60} \approx 2.783 \approx 2.8 \end{aligned} $$ ## Calculate the sum of squared deviations Using the *Sample Standard Deviation* knowledge point $$ \begin{aligned} \sum x^2 \cdot f &= (1^2 \cdot 4) + (2^2 \cdot 20) + (3^2 \cdot 21) + (4^2 \cdot 15) \ &= (1 \cdot 4) + (4 \cdot 20) + (9 \cdot 21) + (16 \cdot 15) \ &= 4 + 80 + 189 + 240 = 513 \end{aligned} $$ ## Calculate the sample variance and standard deviation Using the *Sample Standard Deviation* knowledge point $$ \begin{aligned} s^2 &= \frac{\sum x^2 \cdot f - \frac{(\sum x \cdot f)^2}{n}}{n - 1} \ &= \frac{513 - \frac{167^2}{60}}{59} \ &= \frac{513 - \frac{27889}{60}}{59} \ &= \frac{513 - 464.8167}{59} \ &= \frac{48.1833}{59} \approx 0.81667 \ s &= \sqrt{0.81667} \approx 0.9037 \end{aligned} $$ Rounding to one decimal place gives $s \approx 0.9$. The sample standard deviation is s = $0.9$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Sample Standard Deviation" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given frequency distribution

Calculate the sample size and sum

Calculate the sum of squared deviations

Calculate the sample variance and standard deviation

Answer: