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------------------------------------------------ **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.

\(s^2 = 42.5\)

\(s = 6.52\)