given the following data set: \{23, 25, 22, 24, 27, 24, 26, 25, 23, 27\} that fits a normal distribution, calculate the mean and standard deviation. then, estimate the population percentage within 1 standard deviation of the mean.

the mean of the data set is dropdown

the standard deviation of the data set is dropdown

this means 68% of the data falls between dropdown and dropdown

Question

given the following data set: \{23, 25, 22, 24, 27, 24, 26, 25, 23, 27\} that fits a normal distribution, calculate the mean and standard deviation. then, estimate the population percentage within 1 standard deviation of the mean.

the mean of the data set is dropdown

the standard deviation of the data set is dropdown

this means 68% of the data falls between dropdown and dropdown

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ], "current_concepts": [ "Sample Mean", "Sample Standard Deviation", "Empirical Rule" ] } ## Calculate the sample mean $$ \begin{aligned} n &= 10 \ \sum x &= 23 + 25 + 22 + 24 + 27 + 24 + 26 + 25 + 23 + 27 = 246 \ \bar{x} &= \frac{246}{10} = 24.6 \end{aligned} $$ ## Calculate the sample standard deviation $$ \begin{aligned} \sum (x - \bar{x})^2 &= (23-24.6)^2 + (25-24.6)^2 + (22-24.6)^2 + (24-24.6)^2 + (27-24.6)^2 \ &+ (24-24.6)^2 + (26-24.6)^2 + (25-24.6)^2 + (23-24.6)^2 + (27-24.6)^2 \ &= 2.56 + 0.16 + 6.76 + 0.36 + 5.76 + 0.36 + 1.96 + 0.16 + 2.56 + 5.76 \ &= 26.4 \ s &= \sqrt{\frac{26.4}{10-1}} = \sqrt{\frac{26.4}{9}} \approx 1.71 \end{aligned} $$ ## Apply the Empirical Rule for 1 standard deviation $$ \begin{aligned} \text{Lower bound} &= \bar{x} - s = 24.6 - 1.71 = 22.89 \ \text{Upper bound} &= \bar{x} + s = 24.6 + 1.71 = 26.31 \end{aligned} $$ Given the following data set: {23, 25, 22, 24, 27, 24, 26, 25, 23, 27} that fits a normal distribution, calculate the mean and standard deviation. Then, estimate the population percentage within 1 standard deviation of the mean. The mean of the data set is 24.6 The standard deviation of the data set is 1.71. This means 68% of the data falls between 22.89 and 26.31 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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Question

Explanation:

Calculate the sample mean

Calculate the sample standard deviation

Apply the Empirical Rule for 1 standard deviation

Answer: