Question:Piercing rating of fencing safety jackets. A manufacturer produces safety jackets for competitive fencers. These jackets are rated by the minimum force, in newtons, that will allow a weapon to pierce the jacket. When this process is operating correctly, it produces jackets that have ratings with an average of 840 newtons and a standard deviation of 15 newtons. FIE, the international governing body for fencing, requires jackets to be rated at a minimum of 800 newtons. To check whether the process is operating correctly, a manager takes a sample of 50 jackets from the process, rates them, and calculates$\overline{x}$, the mean rating for jackets in the sample. She assumes that the standard deviation of the process is fixed but is worried that the mean rating of the process may have changed.

a. What is the sampling distribution of $\overline{x}$if the process is still operating correctly?

Step by step solution

01

Given Information

Let x be the rating process.

The sample size is 50.

A manufacturer produces jackets with an average 840 newtons and standard deviation 15 newtons.

02

Sampling distribution

A sampling distribution is a statistical probability distribution derived from repeated sampling of a given population. It represents a population's range of probable outcomes for a statistic, like the average or mode of certain variable. The vast majority of the information evaluated by academics are samples rather than populations.

03

Sampling distributions of  x¯

The mean is given by

${\mu }_{\overline{x}}=840$

The s.d is given by

$$\begin{gathered} {\sigma }_{\overline{X}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{15}{\sqrt{50}} \\ =2.121 \end{gathered}$$

Therefore, the mean and sd are 840 and 2.121.

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