Question

A company produces cans of stewed tomatoes with an advertised weight of 14oz. The standard deviation of the weights is known to be 0.4oz. A quality - control engineer selects a can of stewed tomatoes at random and finds its net weight to be 17.28oz.

Part (a): Estimate the relative standing of that can of stewed tomatoes, assuming the true mean weight is 14oz. Use the z-score and Chebyshev's rule.

Part (b): Does the quality - control engineer have reason to suspect that the true mean weight of all cans of stewed tomatoes being produced is not 14oz? Explain your answer.

Short answer

Part (a): At least 98.51% of the weights lie within 8.2 standard deviations to either side of the mean.

Part (b): The quality - control engineer have to suspected that the true mean weight of all cans of stewed tomatoes is not being produced.

The z-score for the net weight 17.28 of stewed tomatoes lies above 3 standard deviation, that means it is very far from three sigma limits.

Step-by-step solution

Part (a) Step 1. Given information.

Consider the given question,

Weight of stewed tomatoes is 14oz.

The standard deviation of the weights is known to be 0.4oz.

Part (a) Step 2. Determine the relative standing of that can of stewed tomatoes.

Assuming the true mean weight is 17.28oz.

The z-score for stewed tomatoes is given below,

$$\begin{gathered} z=\frac{\overline{x}-\mu }{\sigma } \\ =\frac{17.28-14}{0.4} \\ =8.2 \end{gathered}$$

By applying Chebyshev's rule to the z-score is given below,

$$\begin{gathered} =100\left(1-\frac{1}{8.2^2}\right) \\ =100\left(1-\frac{1}{67.24}\right) \\ =100\left(0.98513\right) \\ =98.513 \end{gathered}$$

Thus, we conclude that at least 98.51% of the weights lie within 8.2 standard deviations to either side of the mean.

Part (b) Step 1. Explain the answer.

The quality - control engineer have to suspect that the true mean weight of all cans of stewed tomatoes being produced is not. Because the z-score for the net weight 17.28 of stewed tomatoes lies above 3 standard deviation, that means it is very far from three sigma limits.