Final answer:
The 95% tolerance limits for a distribution that is bell-shaped and symmetric are typically the mean plus and minus two standard deviations. In this case, the limits are (0.05 cm, 0.25 cm), and the standard deviation is 0.05 cm.
Step-by-step explanation:
Constructing 95% Tolerance Limits for a Distribution
To construct the 95% tolerance limits for the distribution of diameters that contain 95% of the distribution, we will assume that the distribution is bell-shaped and symmetric as suggested by the empirical rule. This rule states that approximately 95% of the data is within two standard deviations of the mean for such a distribution.
To find these tolerance limits, which are a type of interval estimate, we would proceed as follows:
- Calculate the mean of the diameter distribution, which is already provided as 0.15 cm.
- Compute the standard deviation, which would be half the range (0.20 cm - 0.10 cm) / 2 = 0.05 cm.
- To contain 95% of the distribution, we will use the empirical rule. The 95% tolerance limits are the mean plus and minus two standard deviations. So, the lower bound would be 0.15 cm - 2(0.05 cm) = 0.05 cm, and the upper bound would be 0.15 cm + 2(0.05 cm) = 0.25 cm.
- To express these limits, we would say that the tolerance limits are (0.05 cm, 0.25 cm).
Note that if the actual distribution is significantly different from a bell-shaped and symmetric pattern, these calculated limits might not be correct. In practice, we would verify the distribution's shape before applying the empirical rule.
The range referred to in part b of the question is the total spread of the distribution and would be directly the range provided, which is again 0.10 cm to 0.20 cm.
The standard deviation we've calculated for the bell-shaped, symmetric distribution is 0.05 cm, which directly answers part d of the question.