Question

The Ball Corporation's beverage can manufacturing plant in Fort Atkinson, Wisconsin, uses a metal supplier that provides metal with a known thickness standard deviation σ = .000786 mm. Assume a random sample of 55 sheets of metal resulted in an x¯ = .3307 mm. Calculate the 98 percent confidence interval for the true mean metal thickness. (Round your answers to 4 decimal places.)

The 98% confidence interval is from ____ to _____

Answer 1

We know that the 98% confidence interval for population mean is given by:

Answer 2

Given:

• Sample mean ($\bar{x}$): $0.3307\text{mm}$

• Population standard deviation ($\sigma$): $0.000786\text{mm}$

• Sample size ($n$): 55

• Confidence level: 98%

• 
  

Step 1: Find the Critical Z-Value ($z_{\alpha \mathrm{/}2}$)

For a 98% confidence interval, the area in each tail is $1\mathrm{\%}$.
Using the standard normal table:

$$z_{\alpha \mathrm{/}2}=z_{0.01}\approx 2.326$$

Step 2: Calculate the Margin of Error (E)

$$E=z_{\alpha \mathrm{/}2}\times \frac{\sigma }{\sqrt{n}}=2.326\times \frac{0.000786}{\sqrt{55}}$$$$E\approx 2.326\times 0.000106=0.000246$$

Step 3: Compute the Confidence Interval

$$\text{Lower bound}=\bar{x}-E=0.3307-0.000246=0.3305\text{mm}$$$$\text{Upper bound}=\bar{x}+E=0.3307+0.000246=0.3309\text{mm}$$

Answer (Rounded to 4 Decimal Places):

The 98% confidence interval for the true mean metal thickness is from

$$\boxed{{\displaystyle 0.3305}}\text{mm}\text{to}\boxed{{\displaystyle 0.3309}}\text{mm}$$