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Answer :
To solve this problem, we're looking to find a point estimate for the population standard deviation based on a sample of rivet diameters. Here's how you can approach it step by step:
1. List the Sample Data:
The diameters of the rivets are: 6.25, 6.25, 6.29, 6.36, 6.33, 6.36 millimeters.
2. Calculate the Sample Mean:
The sample mean is the average of these measurements. Add all the measured diameters and divide by the number of measurements:
[tex]\[
\text{Sample Mean} = \frac{6.25 + 6.25 + 6.29 + 6.36 + 6.33 + 6.36}{6} = 6.307
\][/tex]
3. Calculate the Sample Variance:
The sample variance uses the squared difference between each measurement and the sample mean, averaged over the number of observations minus one (n - 1, where n is the sample size).
[tex]\[
\text{Sample Variance} = \frac{(6.25 - 6.307)^2 + (6.25 - 6.307)^2 + (6.29 - 6.307)^2 + (6.36 - 6.307)^2 + (6.33 - 6.307)^2 + (6.36 - 6.307)^2}{5}
\][/tex]
4. Calculate the Sample Standard Deviation:
The sample standard deviation is the square root of the sample variance. It serves as a point estimate for the population standard deviation.
[tex]\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}} = 0.051
\][/tex]
5. Round the Result:
Round the sample standard deviation to three decimal places, which is 0.051.
The point estimate for the population standard deviation based on this sample of rivet diameters is 0.051 millimeters.
1. List the Sample Data:
The diameters of the rivets are: 6.25, 6.25, 6.29, 6.36, 6.33, 6.36 millimeters.
2. Calculate the Sample Mean:
The sample mean is the average of these measurements. Add all the measured diameters and divide by the number of measurements:
[tex]\[
\text{Sample Mean} = \frac{6.25 + 6.25 + 6.29 + 6.36 + 6.33 + 6.36}{6} = 6.307
\][/tex]
3. Calculate the Sample Variance:
The sample variance uses the squared difference between each measurement and the sample mean, averaged over the number of observations minus one (n - 1, where n is the sample size).
[tex]\[
\text{Sample Variance} = \frac{(6.25 - 6.307)^2 + (6.25 - 6.307)^2 + (6.29 - 6.307)^2 + (6.36 - 6.307)^2 + (6.33 - 6.307)^2 + (6.36 - 6.307)^2}{5}
\][/tex]
4. Calculate the Sample Standard Deviation:
The sample standard deviation is the square root of the sample variance. It serves as a point estimate for the population standard deviation.
[tex]\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}} = 0.051
\][/tex]
5. Round the Result:
Round the sample standard deviation to three decimal places, which is 0.051.
The point estimate for the population standard deviation based on this sample of rivet diameters is 0.051 millimeters.
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