High School

**Standard Deviation and Variance**

1. For each calculation, use the symbol for the measure you are calculating. For example, if you're calculating a sample mean, use the symbol \(\bar{x}\) when referring to your answer. When referring to the variance, use the symbols \(s^2\) and \(\sigma^2\).

- Do not use your calculator for this one. Write out the formulas and show your work. Round your answers to 3 decimal places.
- (You'll be able to use your calculator on the problems after this one.)
- (4 points)

You're given a sample with \(n = 8\) measurements: \(3, 1, 5, 6, 4, 4, 3, 5\).

A. Calculate the range \((\text{maximum value} - \text{minimum value})\).

\(\text{Range} = 5 - 1 = 4\)

B. Calculate the sample mean.

\(\bar{x} = \frac{3 + 1 + 5 + 6 + 4 + 4 + 3 + 5}{8} = 3.875 \approx 3.9\)

C. Calculate the sample variance and sample standard deviation.

\(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 2.4\)

\(s = \sqrt{s^2} = 1.6\)

D. The range is approximately how many sample standard deviations?

\(\text{Approximately} \, \frac{4}{1.6} \approx 2.5\)

2. Consider the following set of data, which represents a simple random sample of heights (in inches) of full-grown corn plants in Elephanteye town, WI: \(65.8, 62.1, 61.5, 66.6, 65.1, 60.3, 70.2, 69.1, 68.3, 62.1, 65.1, 63.7, 68.5, 71.4, 65.9, 63.8, 66.7, 63.3, 71.9, 56.8\). Round your answers to 2 decimal places.

- (4 points)

A. Calculate the range.

\(\text{Range} = 71.9 - 56.8 = 15.1\)

B. Calculate the statistic that gives an estimate of the population mean.

\(\bar{x} = \frac{\sum x_i}{n} = 65.4\)

C. Calculate sample statistics to estimate the population standard deviation and variance of all full-grown corn plants in Elephanteye town, WI.

\(s^2 = 14.92\)

\(s = 3.86\)

3. Consider the following set of data, which represents a simple random sample of heights (in inches) of full-grown corn plants in Horseye town, WI: \(78.5, 88.8, 76.2, 73.0, 79.5, 66.9, 72.4, 86.6, 67.6, 73.5, 72.6, 72.6, 80.0, 69.1, 79.1, 73.9, 72.3, 83.3, 64.3, 69.1\). Round your answers to 2 decimal places.

- (4 points)

A. Calculate the range.

\(\text{Range} = 88.8 - 66.5 = 22.3\)

B. Calculate the sample mean.

\(\bar{x} \approx 74.97\)

C. Calculate sample statistics to estimate the population standard deviation and variance of all full-grown corn plants in Horseye town, WI.

\(s^2 = 42.5\)

\(s = 6.52\)

Answer :

A. The range is 88.8 - 64.3 = 24.5 (rounded to 1 decimal place).

B. The sample mean is calculated by adding up all the values in the sample and dividing by the number of values in the sample. In this case, the sample mean is (78.5 + 88.8 + 76.2 + 73.0 + 79.5 + 66.9 + 72.4 + 86.6 + 67.6 + 73.5 + 72.6 + 72.6 + 80.0 + 69.1 + 79.1 + 73.9 + 72.3 + 83.3 + 64.3 + 69.1)/20 = 74.97 (rounded to 2 decimal places).

How do we determine the sample variance?

C. To calculate the sample variance, subtract the mean from each value, square the difference, add up the squared differences, and divide by the number of values in the sample minus 1. In this case, the sample variance is

[(78.5 - 74.97)^2 + (88.8 - 74.97)^2 + (76.2 - 74.97)^2 + (73.0 - 74.97)^2 + (79.5 - 74.97)^2 + (66.9 - 74.97)^2 + (72.4 - 74.97)^2 + (86.6 - 74.97)^2 + (67.6 - 74.97)^2 + (73.5 - 74.97)^2 + (72.6 - 74.97)^2 + (72.6 - 74.97)^2 + (80.0 - 74.97)^2 + (69.1 - 74.97)^2 + (79.1 - 74.97)^2 + (73.9 - 74.97)^2 + (72.3 - 74.97)^2 + (83.3 - 74.97)^2 + (64.3 - 74.97)^2 + (69.1 - 74.97)^2]/(20-1) = 42.5 (rounded to 2 decimal places).

The sample standard deviation is the square root of the sample variance. In this case, the sample standard deviation is sqrt(42.5) = 6.52 (rounded to 2 decimal places).

learn more about standard deviation: https://brainly.com/question/475676

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