Questions: An auto transmission manufacturer receives ball bearings from two different suppliers. The ball bearings must have a specified diameter of 16.30 mm with a tolerance of ± 0.1 mm. Recent shipments from the two suppliers had ball bearings with the following diameters. Complete parts (a) through (c). Supplier A: 16.23 16.27 16.32 16.33 16.37 16.41 16.44 Supplier B: 16.19 16.21 16.24 16.35 16.39 16.42 16.45 a. Find the mean and standard deviation for each of the two data sets. Find the mean and standard deviation for the diameters of the ball bearings from Supplier A. mean = s = (Round to the nearest hundredth as needed.)

An auto transmission manufacturer receives ball bearings from two different suppliers. The ball bearings must have a specified diameter of 16.30 mm with a tolerance of ± 0.1 mm. Recent shipments from the two suppliers had ball bearings with the following diameters. Complete parts (a) through (c). Supplier A: 16.23 16.27 16.32 16.33 16.37 16.41 16.44 Supplier B: 16.19 16.21 16.24 16.35 16.39 16.42 16.45 a. Find the mean and standard deviation for each of the two data sets. Find the mean and standard deviation for the diameters of the ball bearings from Supplier A. mean = s = (Round to the nearest hundredth as needed.)
Transcript text: An auto transmission manufacturer receives ball bearings from two different suppliers. The ball bearings must have a specified diameter of 16.30 mm with a tolerance of $\pm 0.1 \mathrm{~mm}$. Recent shipments from the two suppliers had ball bearings with the following diameters. Complete parts (a) through (c). \begin{tabular}{llllllll} Supplier A: & 16.23 & 16.27 & 16.32 & 16.33 & 16.37 & 16.41 & 16.44 \\ Supplier B: & 16.19 & 16.21 & 16.24 & 16.35 & 16.39 & 16.42 & 16.45 \end{tabular} a. Find the mean and standard deviation for each of the two data sets. Find the mean and standard deviation for the diameters of the ball bearings from Supplier A. \[ \begin{array}{l} \text { mean }=\square \\ \mathrm{s}=\square \end{array} \] (Round to the nearest hundredth as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Mean for Supplier A

The mean diameter of the ball bearings from Supplier A is calculated as follows:

\[ \mu_A = \frac{\sum_{i=1}^N x_i}{N} = \frac{114.37}{7} = 16.34 \]

Step 2: Calculate the Standard Deviation for Supplier A

The variance and standard deviation for Supplier A are calculated using the following formulas:

\[ \sigma^2_A = \frac{\sum (x_i - \mu_A)^2}{n-1} = 0.01 \]

Thus, the standard deviation is:

\[ \sigma_A = \sqrt{0.01} = 0.07 \]

Step 3: Calculate the Mean for Supplier B

The mean diameter of the ball bearings from Supplier B is calculated as follows:

\[ \mu_B = \frac{\sum_{i=1}^N x_i}{N} = \frac{114.25}{7} = 16.32 \]

Step 4: Calculate the Standard Deviation for Supplier B

The variance and standard deviation for Supplier B are calculated using the following formulas:

\[ \sigma^2_B = \frac{\sum (x_i - \mu_B)^2}{n-1} = 0.01 \]

Thus, the standard deviation is:

\[ \sigma_B = \sqrt{0.01} = 0.11 \]

Final Answer

For Supplier A:

  • Mean: \( \mu_A = 16.34 \)
  • Standard Deviation: \( \sigma_A = 0.07 \)

For Supplier B:

  • Mean: \( \mu_B = 16.32 \)
  • Standard Deviation: \( \sigma_B = 0.11 \)

The final answers are: \[ \boxed{\mu_A = 16.34, \sigma_A = 0.07} \] \[ \boxed{\mu_B = 16.32, \sigma_B = 0.11} \]

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