Question

A sample of 6 sheets of cardstock is randomly selected and the following thicknesses are measured in millimeters. Give a point estimate for the population standard deviation. Round your answer to three decimal places. 2.88,2.91,2.97,2.97,2.92,2.87

Answer 1

To estimate the population standard deviation from a sample, calculate the sample standard deviation using the sample data by finding the mean, deviations, squared deviations, variance, and then take the square root of the variance. Round the final result to three decimal places to get the point estimate.

Step-by-step explanation:

To give a point estimate for the population standard deviation, we first need to calculate the sample standard deviation and then use it as our best estimate for the population standard deviation. The steps for calculating the sample standard deviation are as follows:

1. Calculate the sample mean (average).
2. Subtract the sample mean from each measurement to find the deviation of each measurement.
3. Square each deviation to get the squared deviations.
4. Sum up all the squared deviations.
5. Divide the sum by the sample size minus one (n-1) to get the variance.
6. Take the square root of the variance to find the sample standard deviation.

Now, let's calculate it for the given data of cardstock thicknesses: 2.88, 2.91, 2.97, 2.97, 2.92, 2.87.

1. Sample mean: (2.88+2.91+2.97+2.97+2.92+2.87) / 6 = 17.52 / 6 = 2.92 mm
2. Squared deviations: Sigma((x_i - mean)^2), for each thickness x_i.
3. Sum of squared deviations: Sigma((x_i - 2.92)^2).
4. Variance: (Sum of squared deviations) / (6-1).
5. Sample standard deviation: sqrt(Variance).

Answer 2

Standard deviation = 0.039 mm

Step-by-step explanation:

The standard deviation is the square root of variance. And variance is an average of the squared deviations from the mean.

Mathematically,

Standard deviation = σ = √[Σ(x - xbar)²/N]

x = each variable

xbar = mean

N = number of variables = 6

We first calculate the mean

xbar = (Σx)/N

xbar = (2.88+2.91+2.97+2.97+2.92+2.87)/6

xbar = (17.52/6) = 2.92 mm

Σ(x - xbar)² = [(2.88 - 2.92) + (2.91 - 2.92)² + (2.97 - 2.92)² + (2.97 - 2.92)² + (2.92 - 2.92)² + (2.87 - 2.92)²] = 0.0092

σ = √[Σ(x - xbar)²/N] = √(0.0092/6) = √0.0015333

σ = 0.039 mm